Assessment 3

library(cluster)
library(tidyverse)

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library(ggplot2)
library(dplyr)
library(rattle)

Loading required package: bitops
Rattle: A free graphical interface for data science with R.
Version 5.5.1 Copyright (c) 2006-2021 Togaware Pty Ltd.
Type 'rattle()' to shake, rattle, and roll your data.

library(kableExtra)

Attaching package: 'kableExtra'

The following object is masked from 'package:dplyr':

    group_rows

library(rpart)
library(rpart.plot)

Rows: 850 Columns: 9
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (2): renewed, gender
dbl (7): id, num_contacts, contact_recency, num_complaints, spend, lor, age

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

Part A - Exploratory Analysis of the Music Subscription Dataset

Renewels by Gender

As we observe the data within the barchart, it is broken down by gender, and whether they renewed their subscription or not. We notice that females have a higher rate of cancelation on their subscriptions, while men, have a higher rate of renewing their subscription. However, we also observe tat females hold a significant amount less than males in overall subscriptions

Renewels by Number of Contacts

Customers who renewed tend to have slightly more contact with the company, as shown by the higher median number of contacts who did renew their subscription. However, the distributions overlap. Both groups show a long tail of customers with high contact counts, suggesting that a few customers interacted excessivley regardless of their renewal of their subscription. This would indicate that the number of contacts alone is not a strong indication of whether the customer will renew.

Contact Recency by Renewal Status

Both customers who renew, and who don’t renew display similar distributions of contact recency. The medians look very similar, and the density shapes overlap across the entire range. This would suggest that how recent a customer has contacted the brand, does not directly affect or influence their renewel. The heavy concentration of values around the same recency level for both groups reinforces that this variable has little predictive value for churn.

Customaer age and Renewal Outcome

As we can see from the graph above, it appears to show that ages impacts renewal minimally. Both younger and older customers appear in both renewal categories, and the jitter points highlight the lack of seperation between the groups. This would suggest that age is unlikely to be a key driver of subscription renewal.

Length of Relationship by Renewal Status

Customers who renew show a slightly longer average relationship duration, with a visibly higher median and more density in the upper range. However, the distributions do still overlap, meaning many long-term customers churned while many short-term have renewed. This would suggest that although there is a weak tendecny for longer customers to renew, relationship length is not a strong predictor of renewal behaviour.

Customer Complaints by Renewal Status

The majority of customers regardless of their renewal status, have little to none complaints, which explains the large spike at 0-1. As the number of complaints increase, the number of customers drops drastically for both renewed and not renewed. There is no clear pattern saying that customers with more complaints are more likely to renew, largely because high complaints seem to occur so infrequently.

Part B - Predicting Customers Renewal

Rows: 150 Columns: 9
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (2): renewed, gender
dbl (7): id, num_contacts, contact_recency, num_complaints, spend, lor, age

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
Rows: 850 Columns: 9
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (2): renewed, gender
dbl (7): id, num_contacts, contact_recency, num_complaints, spend, lor, age

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

Rows: 150 Columns: 9
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (2): renewed, gender
dbl (7): id, num_contacts, contact_recency, num_complaints, spend, lor, age

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

            lor           spend             age    num_contacts contact_recency 
     21.5220728      10.3707565       9.9887537       3.7116877       0.5951338 

Call:
rpart(formula = renewed ~ num_contacts + contact_recency + num_complaints + 
    spend + lor + gender + age, data = subtrain_data)
  n= 850 

          CP nsplit rel error    xerror       xstd
1 0.19339623      0 1.0000000 1.0778302 0.03428302
2 0.01179245      1 0.8066038 0.9245283 0.03427681
3 0.01000000      4 0.7617925 0.8396226 0.03392448

Variable importance
            lor           spend             age    num_contacts contact_recency 
             47              22              22               8               1 

Node number 1: 850 observations,    complexity param=0.1933962
  predicted class=No   expected loss=0.4988235  P(node) =1
    class counts:   426   424
   probabilities: 0.501 0.499 
  left son=2 (466 obs) right son=3 (384 obs)
  Primary splits:
      lor             < 139.5 to the left,  improve=16.323670, (0 missing)
      age             < 60.5  to the left,  improve=13.784490, (0 missing)
      spend           < 182   to the left,  improve=12.232030, (0 missing)
      contact_recency < 7.5   to the right, improve= 5.498858, (0 missing)
      num_contacts    < 3.5   to the left,  improve= 5.307063, (0 missing)
  Surrogate splits:
      age             < 60.5  to the left,  agree=0.732, adj=0.406, (0 split)
      spend           < 422   to the left,  agree=0.684, adj=0.299, (0 split)
      contact_recency < 8.5   to the right, agree=0.565, adj=0.036, (0 split)
      num_contacts    < 2.5   to the left,  agree=0.560, adj=0.026, (0 split)

Node number 2: 466 observations,    complexity param=0.01179245
  predicted class=No   expected loss=0.4098712  P(node) =0.5482353
    class counts:   275   191
   probabilities: 0.590 0.410 
  left son=4 (82 obs) right son=5 (384 obs)
  Primary splits:
      spend           < 182   to the left,  improve=5.482157, (0 missing)
      lor             < 42.5  to the left,  improve=5.434363, (0 missing)
      age             < 61.5  to the left,  improve=4.067404, (0 missing)
      num_contacts    < 7.5   to the left,  improve=3.721617, (0 missing)
      contact_recency < 7.5   to the right, improve=2.971256, (0 missing)
  Surrogate splits:
      lor < 21    to the left,  agree=0.987, adj=0.927, (0 split)

Node number 3: 384 observations
  predicted class=Yes  expected loss=0.3932292  P(node) =0.4517647
    class counts:   151   233
   probabilities: 0.393 0.607 

Node number 4: 82 observations
  predicted class=No   expected loss=0.2439024  P(node) =0.09647059
    class counts:    62    20
   probabilities: 0.756 0.244 

Node number 5: 384 observations,    complexity param=0.01179245
  predicted class=No   expected loss=0.4453125  P(node) =0.4517647
    class counts:   213   171
   probabilities: 0.555 0.445 
  left son=10 (356 obs) right son=11 (28 obs)
  Primary splits:
      num_contacts   < 7.5   to the left,  improve=3.286592, (0 missing)
      age            < 61    to the left,  improve=3.189001, (0 missing)
      gender         splits as  LR,        improve=2.683644, (0 missing)
      num_complaints < 1.5   to the left,  improve=2.393375, (0 missing)
      lor            < 45.5  to the left,  improve=1.671696, (0 missing)
  Surrogate splits:
      lor < 137.5 to the left,  agree=0.93, adj=0.036, (0 split)

Node number 10: 356 observations,    complexity param=0.01179245
  predicted class=No   expected loss=0.4269663  P(node) =0.4188235
    class counts:   204   152
   probabilities: 0.573 0.427 
  left son=20 (329 obs) right son=21 (27 obs)
  Primary splits:
      age            < 61    to the left,  improve=3.357262, (0 missing)
      gender         splits as  LR,        improve=2.949544, (0 missing)
      num_complaints < 1.5   to the left,  improve=1.278902, (0 missing)
      lor            < 42.5  to the left,  improve=1.232817, (0 missing)
      spend          < 403   to the right, improve=1.196010, (0 missing)

Node number 11: 28 observations
  predicted class=Yes  expected loss=0.3214286  P(node) =0.03294118
    class counts:     9    19
   probabilities: 0.321 0.679 

Node number 20: 329 observations
  predicted class=No   expected loss=0.4072948  P(node) =0.3870588
    class counts:   195   134
   probabilities: 0.593 0.407 

Node number 21: 27 observations
  predicted class=Yes  expected loss=0.3333333  P(node) =0.03176471
    class counts:     9    18
   probabilities: 0.333 0.667 

   id renewed num_contacts contact_recency num_complaints spend lor gender age
1 187      No            0              28              0   213 248   Male  45
2 269      No            1              12              2   425  82   Male  60
3 376      No            0              28              2     0  15 Female  53
4 400      No            1              11              1     0  12   Male  44
5 679     Yes            0              28              0   216 300   Male  68
6 565     Yes            0              28              0   425 349 Female  68
         No       Yes train_preds
1 0.3932292 0.6067708         Yes
2 0.5927052 0.4072948          No
3 0.7560976 0.2439024          No
4 0.7560976 0.2439024          No
5 0.3932292 0.6067708         Yes
6 0.3932292 0.6067708         Yes

      Predicted
Actual  No Yes
   No  257 169
   Yes 154 270

      Predicted
Actual No Yes
   No  36  38
   Yes 35  41

[1] 0.5133333

      Predicted
Actual  No Yes
   No  266 160
   Yes 172 252

[1] 0.6094118

      Predicted
Actual No Yes
   No  41  33
   Yes 38  38

[1] 0.5266667

[1] 0.5266667

[1] 150

 Factor w/ 2 levels "No","Yes": 2 1 1 2 2 1 1 1 2 1 ...

 No Yes 
 74  76 

Interpretation

A clear rule noted from our pruned classification tree is if a customers spend is at least €182, the model predicts that the customer will renew. This split appears at the top of the tree, and customers who fall into this group show a majority of ‘Yes’ outcomes. The node is not extremely pure, but it is pure enough for the model to confidently classify this segment as likely renewers because most people with higher spend levels do in fact renew.

A strong rule for churn is if a customer spends less than €182 and has fewer than 7.5 contacts with the company, the model predicts they will churn. This node is the purest churn group in the tree and the level of dominance makes the node highly pure and signals that low-spend, and low contact customers behave consistently and leave.

The pruned model keeps only the variables that help distinguish between renewers and chruners, which are spend, length of relationship, and number of contacts. Spend is the most important because it forms the root split and produces the biggest reduction in uncertanity. Length of relationship helps seperate custoemrs further within the higher-spend group, while number of contacts refrines the prediction among lower-spend customers. Variables such as age, gender, contact recency, and number of complaints are removed during pruning, which means they don’t provide enough predictive value once the stronger variables are considered.

Comparison of Visual Exploration and Classfication Tree

The visual exploration conducted in Part A suggested that variables such as age, contact recency, number of complaints, and length of relationship might just influence behaviour, as differences were observed between customers who renew, and customers who churn. However, when the classification tree was applied, many variables did were classed as unimportant. the final model relied mainly on spend, length of relationship, and number of contacts, showing that these variables gave the greatest clarity. Variables like age, gender, contact recency, and number of complaints appeard interesting, however, they did not improve the prediction once included in the tree. This would suggest that visual patterns do not always translate in strong predictors, while variables such as spend, which appeard less dominant visually, was in fact important in a the tree.

Accuracy Assessment and Overfitting

The unpruned classification tree performs better on the training data than on the testing data, with training accuracy around 60% and testing accuracy dropping to 51%. This gap shows that the model learned patterns specific to the training data that do not suit new customers. This difference in performance is clear evidence of overfitting

Pruned Classification Tree and Overfitting Reduction

A pruned classification tree was then created with a maximum depth of 3 to reduct complexity. Pruning simplified the model by removing the weaker splits in the tree, while keeping the most important variables. As a result, the gap between the training and testing data suggests that the pruned tree overfits less than the original model. Although the overall accuracy did not improve much, the pruned three is more reliable because it adapts better to unseen data. The pruned tree had an accuracy of 53% on the training data, while having 52% on the testing data.

Marketing Actions Based off Analysis

The classification tree shows that spend, LOR, and number of contcts are important variables of renewal. The company should prioritise retention campaigns for low-spending customers with few interactions, as this group is most likely to churn. Encouraging engagement through targeted emails, or personalised offers could improve the renewal rates. High-spending and long-term customers should be rewarded with loyalty incentives, as they are already more likely to renew their subscription. The model can be used for marketing byy segmenting customers based on their predicted renewal likelihood and focusing retention efforts where they will have bigger impacts.

Part C - Segmenting Consumers Based on Energy Drink Preference

Rows: 840 Columns: 8
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (3): ID, Gender, Age
dbl (5): D1, D2, D3, D4, D5

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

            1         2         3         4         5         6         7
2   1.7298913                                                            
3   1.0414878 2.5854323                                                  
4   2.3376325 2.4383106 2.5154602                                        
5   0.4491527 2.0003037 1.1342112 2.2940766                              
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99  2.0689960 2.2002343 2.0829333 1.4549339 2.2104205 2.0265761 1.4528659
100 2.6596735 3.0590945 2.6105817 3.5034006 2.8429833 2.2790209 2.2914097
101 3.6843374 4.0593020 3.5270277 2.9422285 3.6568571 3.2891828 2.0761999
102 3.3896816 3.9727991 2.9943151 2.2765560 3.3597921 2.9553888 2.2372766
103 2.2022828 2.7059153 2.0563445 1.6006835 2.2476182 1.9992380 1.5645881
104 2.2326288 3.1054258 1.8929716 3.0979726 2.3642855 1.5573507 1.9635525
105 1.9036743 2.9026268 1.5389299 1.8111283 1.8499291 1.4617472 1.4002269
106 3.3629428 3.4806455 3.2204494 3.4442807 3.5667284 2.9580557 2.5510775
107 3.2605089 3.7198723 2.9727045 2.6693477 3.2913001 2.7710957 1.8443054
108 3.7257113 3.5884575 3.7268043 3.6640832 3.9106383 3.4357925 2.6286555
109 1.4075636 1.9060594 1.7781537 2.6955097 1.6082443 1.4155670 1.6072884
110 2.3549647 2.7397330 2.3832307 3.6789733 2.6378070 2.0145937 2.5040106
111 2.2494341 3.3992028 1.7318013 2.6851118 2.2041360 1.5180038 1.7257719
112 3.0872694 3.4158920 2.6970513 3.0720670 3.3080809 2.5650118 2.4929961
113 1.8874775 2.8902555 1.2933394 3.0726850 2.1380510 1.2004693 2.4552663
114 2.7314283 3.7217317 2.0803516 3.3130242 2.8400555 1.9060594 2.4473782
115 3.0069219 3.1106453 2.8765367 3.0739472 3.2332254 2.6676699 2.4084779
116 3.0903246 3.1399009 3.3200795 2.5774210 3.0575101 2.9897370 1.2365475
117 3.8769363 3.4334995 4.0382085 3.0784157 4.0049127 3.7713031 2.3176635
118 3.0372658 2.8137745 2.9408631 2.2104982 3.2614645 2.9012197 2.3348263
119 2.9497419 2.9774145 2.8247213 2.9175222 3.1801170 2.6117140 2.1574759
120 1.6540620 3.1186699 1.2270309 3.0100320 1.5919117 0.9004469 2.0339304
            8         9        10        11        12        13        14
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9   1.4075636                                                            
10  2.4798301 2.0557823                                                  
11  2.8765367 2.2790456 1.0178348                                        
12  1.4705011 1.8431487 2.3918437 2.7640523                              
13  2.9995983 2.9921975 1.7207099 1.3155002 2.8606159                    
14  2.4034246 2.0490707 2.0751122 1.4851893 2.7560638 1.7020202          
15  2.1219552 1.7687489 1.1202279 1.6215795 1.5356921 2.1048256 2.3386922
16  3.2746647 3.2430119 1.9870751 1.6746806 2.8409839 1.1117433 2.2054997
17  2.5391969 1.8116369 1.4528659 1.0470754 2.1010539 1.8267451 1.5984134
18  1.0416064 0.7610549 2.3198492 2.5026224 1.3082827 2.9234945 2.0915112
19  1.5690338 0.7693385 2.3225800 2.4109326 1.7429337 2.9995571 2.0945396
20  2.6516241 2.4980702 2.3471348 2.7614760 1.3754623 3.0443500 3.3535053
21  2.5115933 2.2403032 0.3981752 1.2294788 2.3584683 1.6740068 2.2580540
22  1.9587948 1.8639031 1.4719263 2.3998469 2.0043826 2.8919144 2.9804789
23  2.7340842 2.2210532 1.5573507 1.9115014 2.4163543 2.4817990 2.6970513
24  2.2113391 1.5984461 1.9152859 2.2504366 1.7602399 2.8724583 2.6752938
25  2.8999242 3.1000023 2.4573297 2.0003037 2.7434202 1.0128552 1.6573137
26  1.4922957 1.6863972 1.8599492 1.8288502 2.0829333 1.7867407 1.1444998
27  2.3117123 2.3466610 1.7707176 1.4397179 2.0915112 1.0115880 1.3082827
28  3.0145152 2.4383106 1.0416064 1.2294788 2.5473070 1.9309840 2.4546300
29  2.5522734 3.0042418 1.8790636 2.4753670 2.1626398 2.1316687 3.0175020
30  2.5391969 2.1331610 1.4528659 1.3155002 1.9442874 1.6440267 1.9553192
31  2.0441311 1.8659914 3.0702881 3.4607205 2.9861699 3.9752369 3.1530578
32  1.2538917 1.1389480 2.4559151 2.9191973 1.2158704 3.3856946 2.8566639
33  1.5480251 1.1313788 2.1917813 2.7744504 1.6657411 3.3882315 2.9234945
34  2.9312627 2.1671955 1.7003980 1.3155941 2.4882547 2.0587109 1.9923938
35  3.0067135 1.8450469 2.4195073 2.1316687 3.0705813 3.1113162 2.2222182
36  2.9420073 2.1339358 1.6751770 1.6746806 3.4484041 2.6282730 2.0567098
37  2.7340842 2.2210532 1.5573507 1.9115014 2.4163543 2.4817990 2.6970513
38  2.0465962 1.8477391 1.4549339 1.8452785 1.7338666 2.1461638 2.3133681
39  2.0587109 1.5389299 1.9020921 1.5032463 2.1941380 1.9152214 0.9133725
40  1.4783678 1.0470754 2.8826134 3.1976872 2.4097928 3.8004975 2.7488573
41  2.1236735 1.3221091 2.6676699 2.8919144 1.6971657 3.5867755 2.9345014
42  1.8111283 1.3240603 2.3266676 2.3564172 2.7888003 2.9039116 1.6981504
43  2.7749976 2.3892585 1.4583381 1.2114344 3.1353319 1.6040434 1.5221097
44  2.2328501 2.0312865 2.0097877 2.1120583 1.5507228 2.2986158 2.3791134
45  1.8349499 1.5283178 1.0470754 1.7550755 2.2680317 2.3376325 2.1625827
46  1.4528659 0.6002347 1.6558743 2.1048256 1.6751770 2.8211328 2.2276554
47  1.2365475 1.0115880 2.0123079 2.0106193 1.4682334 2.3319689 1.4676136
48  3.2204494 2.8752393 1.4854017 1.2878243 2.8034184 1.3963204 2.2975508
49  3.2062459 2.6166450 1.2158704 1.1441284 3.4525124 1.9635525 2.0644723
50  3.1107461 2.7159013 1.3963204 1.0414878 2.5702134 1.2538917 2.0947215
51  3.2548283 2.6713645 1.8853069 1.2270309 2.7050557 1.6392224 1.9350582
52  2.6451406 2.0675205 2.6766312 3.1483188 2.9595046 3.8518842 3.3972700
53  1.2810941 1.1202279 2.3265794 2.9073622 1.5450921 3.4595404 2.9191973
54  2.5597186 1.9152214 0.8076269 0.9064062 2.5448547 1.8011098 1.8287827
55  3.3805932 2.4280425 1.7139129 1.3963204 2.8247213 2.4687382 2.4444305
56  1.4865267 1.6914846 1.5053792 1.5984134 1.7442071 1.5389299 1.3155002
57  2.7336142 1.7717889 2.2105881 2.3457531 2.4387391 3.1975505 2.7823377
58  3.3627119 2.8576018 1.2365475 1.0332048 2.9511527 1.5480251 2.3624357
59  2.7208106 1.9086730 2.0445213 2.4750357 3.0130250 3.3466911 2.9408631
60  2.7277280 1.9521757 0.9236324 1.1202279 2.9191992 2.2104205 1.9725211
61  2.7061435 1.8116369 1.4528659 1.0470754 2.6535828 2.0524836 1.5984134
62  2.4266442 2.6395050 1.4549339 1.5984461 2.5610818 1.0262418 1.8908327
63  2.0829333 2.0761999 1.7554141 1.9202407 2.7351591 2.0445213 1.7832630
64  2.3023508 2.0873989 1.9571561 1.4308192 2.6082888 1.4505707 0.3981752
65  2.3840644 1.8857149 1.7207460 1.3638640 1.9331474 1.7501730 1.5573507
66  2.5115933 2.2403032 0.3981752 1.2294788 2.3584683 1.6740068 2.2580540
67  2.7857329 1.9483337 2.8221138 2.7880516 2.8138504 3.4822258 2.7552166
68  2.5905024 1.7867407 0.9004469 0.8142657 2.6366486 1.9286335 1.6937579
69  1.7338666 1.3076471 1.8776540 2.0941509 2.6044100 2.6523142 1.8011098
70  2.7335689 1.6875278 2.0216672 2.0265761 3.1914822 3.0262534 2.1875287
71  1.7418091 1.6863972 1.0190941 1.3155941 2.0829333 1.5445031 1.4549339
72  2.1236735 1.3221091 1.8877542 1.7418091 1.6971657 2.3998469 1.8116369
73  2.8306810 1.8706564 2.3054282 2.1875287 3.1441856 3.0433585 2.2435265
74  2.0356695 1.4822565 1.2884291 1.1202279 2.0689960 1.7897806 1.2124279
75  2.9003022 1.8877542 2.3023508 2.7853105 2.7061208 3.7875935 3.4113591
76  2.3097023 1.2455510 2.0003037 1.9020018 2.7523933 2.7977648 1.7565458
77  1.1343201 0.6143829 1.9923688 2.2922540 1.2287658 2.8180854 2.1391954
78  3.1855707 2.1166109 2.5320985 2.3281775 2.8467432 3.3243915 2.6117140
79  3.1410624 2.0321700 2.5895233 2.0048327 3.2219086 2.9717083 1.7318013
80  1.7555938 1.5538377 2.2054997 2.2234715 1.3678440 2.5497595 1.9870751
81  3.1566390 2.3754509 2.5381606 1.9478633 3.6341827 2.6886987 1.3592470
82  3.4496629 2.3274890 3.1147014 2.9301321 2.8442661 3.8721835 3.1653392
83  3.5239907 2.5292166 2.6442183 2.7050557 2.6771364 3.6419860 3.4359942
84  3.2459844 3.0388391 2.6437216 2.0216672 3.1052197 1.9010853 1.6657411
85  2.8447469 1.6320556 2.9554101 2.7553753 2.8854117 3.6618894 2.5414208
86  3.5003459 2.9073622 2.0887492 1.1202279 3.2459844 1.5340442 1.5480251
87  4.1338978 3.3423908 3.1838227 2.2312158 3.9395475 2.6959549 2.0756326
88  4.8151250 3.7623056 3.3751042 2.5479587 4.6213833 3.4488973 2.9158269
89  3.9715678 3.4707396 3.5935644 2.8624539 3.9125007 3.1117318 2.2765560
90  2.6442183 2.2079103 2.5997482 2.1494648 2.0812144 2.4108835 1.8277888
91  3.3359041 2.1331610 2.4322547 1.9060594 3.4806455 3.0473071 1.9553192
92  3.1769534 2.5465983 2.9128721 2.6098538 3.1219853 3.2281179 2.3995958
93  2.5740851 1.5176705 2.5553013 2.4713907 2.0945127 3.2891828 2.5499048
94  4.5923336 3.7960809 3.4961222 2.6902131 4.0890828 3.2076250 2.8903040
95  4.3384541 3.5782984 3.6975521 2.8329187 4.1688590 3.2983188 2.4373949
96  2.5553013 1.9211110 1.3939884 1.2158704 2.1120583 1.9635620 1.8349499
97  2.3892209 1.7442071 2.0856237 1.6616028 2.1825769 2.2283133 1.3826348
98  3.5501205 3.0696110 2.3217555 1.7257719 3.4778091 2.0938596 1.9415496
99  2.4108835 1.5348055 1.5661312 1.3155002 2.8315408 2.2778961 1.2761367
100 3.4333884 2.0780387 2.8330459 2.7313089 3.2510344 3.9138626 3.0535043
101 4.1325786 3.4197098 3.4536811 2.5687617 3.9138626 2.9694851 2.1979739
102 3.4949397 3.1000023 3.1374373 2.4297606 3.6378455 2.5948874 1.6573137
103 2.4450333 2.2074066 1.9415496 1.4397179 2.3754509 1.5138224 1.0379931
104 2.8929305 2.0437035 2.5854323 2.3679018 2.3528347 3.1245210 2.4727022
105 2.0853388 1.7356257 2.3516789 2.0003037 2.1757252 2.3106264 1.2158704
106 4.0530424 3.2405982 2.6524532 2.1272836 3.4439198 2.8591498 2.8136615
107 3.6894251 2.7710957 3.0187146 2.2849645 3.6834325 3.0562845 1.9707250
108 4.4637534 3.1706919 3.1035608 2.6292118 4.3091838 3.8643627 3.1832588
109 2.3163875 1.0621387 2.0356695 2.1421565 2.1272836 3.1109336 2.3938057
110 3.2655210 2.2213238 2.4826210 2.5758409 2.5798571 3.5710874 3.2161155
111 2.5905630 1.7832630 2.9568218 2.6465929 2.6084348 3.3108901 2.0761999
112 3.5771777 2.5650118 2.4815495 2.1973515 3.5310039 3.2532455 2.5805300
113 2.3447095 1.5221097 2.3217555 2.5556086 2.2337105 3.3896816 2.7060047
114 3.1493993 2.3204408 3.0266367 2.8128374 2.9404208 3.5878788 2.7059153
115 3.6377119 2.9778928 2.1835071 1.7429337 3.0456021 2.3642855 2.5066034
116 3.7390333 2.8395133 2.9672225 2.1293704 3.5229666 2.7770989 1.9657428
117 4.5465731 3.5314791 2.9050614 2.0431155 4.3685925 2.9574682 2.6112157
118 3.4005221 2.5818003 1.7887773 1.4419266 3.6440292 2.4678608 2.0201256
119 3.5905912 2.4383106 2.2113316 1.8477391 3.4923256 3.0718503 2.4546300
120 2.0945127 1.6006835 2.9382021 2.9012197 1.7233786 3.3995049 2.5213889
           15        16        17        18        19        20        21
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
12                                                                       
13                                                                       
14                                                                       
15                                                                       
16  2.2940766                                                            
17  1.2294788 1.9846402                                                  
18  1.7679150 3.1552452 1.8734397                                        
19  1.7643260 3.3588371 1.7418091 0.6712101                              
20  1.3076471 2.9438525 2.0689960 2.2460246 2.3333325                    
21  1.0470754 2.0265761 1.6082443 2.4201923 2.4228100 2.2435265          
22  1.3383450 3.1093216 2.3496539 2.0766578 2.1882379 2.1011873 1.4170474
23  1.1250933 3.0100320 1.7170984 2.3428044 2.0340019 1.9866735 1.3963204
24  1.0470754 3.2191119 1.6215795 1.6082443 1.2933394 1.5266262 1.8734397
25  2.5026224 1.4809152 2.0887492 2.8211328 2.9744174 3.1976872 2.4248558
26  1.8877542 2.4070673 1.8079878 1.5445031 1.6915576 2.8776635 1.9020921
27  1.7867407 1.2124279 1.3963204 2.1391954 2.3376325 2.5658142 1.8149337
28  1.0470754 2.2435265 1.2884291 2.6045597 2.4228100 2.0265761 0.9624969
29  1.9402267 1.8811528 2.6105817 2.8693583 3.2381834 2.3840644 1.7479272
30  0.9367199 1.9846402 0.7963504 2.0356695 1.9152214 1.7356257 1.3972387
31  2.8600309 4.5859691 3.2480172 2.0825502 1.8706564 3.5860211 3.0959995
32  1.7243896 3.5756524 2.2986158 0.8968765 1.1202279 1.9202407 2.4879836
33  1.6084862 3.6434448 2.3023508 1.2761367 1.2810941 2.0437035 2.2276554
34  1.3963204 2.4561898 0.7804271 2.2461619 1.9152859 2.1941380 1.7463954
35  2.2650718 3.6049344 1.7338666 2.2043353 1.6144595 3.0809781 2.5781259
36  2.3675648 3.2080867 2.1884807 2.6932093 2.4397779 3.6222566 1.8970860
37  1.1250933 3.0100320 1.7170984 2.3428044 2.0340019 1.9866735 1.3963204
38  0.7610549 2.6523142 1.5236535 1.7687489 1.6364443 1.6875278 1.2810941
39  1.7643260 2.4397779 1.1213722 1.4922957 1.3328253 2.6871666 2.0232609
40  2.6117140 4.1577433 2.8147009 1.3076471 1.3221091 3.2223612 3.0169815
41  1.8571888 3.7242763 2.0490707 1.2538917 1.0470754 1.8851104 2.7553753
42  2.4777492 3.4858918 2.3375861 1.6746806 1.5507228 3.5148140 2.4912044
43  2.0829333 2.4736442 1.8706564 2.6330125 2.4636843 3.3514678 1.5117187
44  1.0941189 2.6907566 1.4396997 1.7318013 1.5884767 1.4583381 1.8877542
45  1.4002269 2.7805249 1.9402267 1.8776540 1.8877542 2.5817526 1.1202279
46  1.3155002 3.0718530 1.7093117 1.0128552 1.0190941 2.1272836 1.7937530
47  1.6375035 2.6166450 1.4756215 0.6664127 0.9458482 2.3428044 2.1272079
48  1.5445031 1.9937367 1.5138224 2.8956538 2.7333290 2.4427972 1.3155941
49  2.1941380 2.5054858 2.0366062 3.0323372 2.8865299 3.4677160 1.3978452
50  1.4170474 1.5507228 1.1313788 2.7153582 2.6262698 2.2430309 1.3383450
51  1.7501730 1.8989303 0.9133725 2.6545903 2.4713907 2.4372436 1.9268953
52  2.3411381 4.4244492 2.8923161 2.3705692 1.9635525 2.9861699 2.6468493
53  1.8811528 3.5454505 2.4660400 1.2176447 1.5480251 2.2433964 2.4266442
54  1.2158704 2.3475071 1.2114344 2.2040649 1.9923688 2.4789036 0.9004469
55  1.7318013 2.3605990 1.0941189 2.6920584 2.5115933 2.3711071 1.9313764
56  1.5283986 1.9211110 1.5635480 1.5389299 1.8116369 2.4854863 1.5571480
57  1.6144595 3.5909049 1.7442071 1.9813098 1.4528659 2.1378209 2.2461619
58  1.5584237 1.9238227 1.4505707 3.0007100 2.8443870 2.5154602 1.1706862
59  2.0261824 3.9010917 2.4230293 2.3892578 1.9860747 2.9575323 2.0829333
60  1.7832630 2.4728731 1.6636069 2.4573297 2.3692405 2.9906560 1.2828929
61  1.5450921 2.5623956 0.9357839 2.0941509 1.7418091 2.6282730 1.6082443
62  1.8116369 1.8659914 2.0431155 2.5848249 2.6752938 2.8728727 1.2810941
63  2.0366062 2.9063020 2.2114743 2.1941380 2.1010539 3.2092666 1.7096594
64  2.1625827 2.0948983 1.5480251 2.0532597 2.0563445 3.1837389 2.0751122
65  1.2810941 2.0941509 0.6002347 1.7501730 1.6084862 2.0231761 1.7662135
66  1.0470754 2.0265761 1.6082443 2.4201923 2.4228100 2.2435265 0.0000000
67  2.3054282 4.0594100 2.2234715 2.0526203 1.4002269 2.8556066 2.8500649
68  1.3978452 2.3810361 1.1441284 2.1678004 1.9521757 2.6339477 1.1342112
69  2.0827464 3.2660558 2.1680753 1.6875278 1.5645881 3.1992847 2.0003037
70  2.2652583 3.5980117 2.0825502 2.2680317 1.8384526 3.3608575 2.2090395
71  1.3963204 2.0445213 1.5690338 1.7867407 1.9152859 2.5820616 1.0941189
72  1.2933394 2.6008730 0.8036442 1.2538917 1.0470754 1.8851104 2.0097877
73  2.2749709 3.7108048 2.0368137 2.2527783 1.6875278 3.2494881 2.4063727
74  1.5180038 1.8676260 1.0416064 1.6558743 1.7867407 2.5133541 1.5661312
75  1.9286335 3.9864889 2.3831905 2.3396599 2.0303792 2.3612499 2.4034246
76  2.0504023 3.3380759 1.7338666 1.7338666 1.2961786 3.0809781 2.1895050
77  1.4549339 2.9911338 1.7042775 0.4491527 0.8076269 2.0089639 2.1083557
78  2.4815495 3.0175922 1.8443054 2.4375743 2.4297606 2.9037319 2.8557838
79  2.7219748 3.0730553 1.7196967 2.3791134 2.1786366 3.5103151 2.9068211
80  1.8278773 2.3319689 1.6144595 1.2961786 1.7338666 2.1712104 2.3784336
81  3.0656537 2.9525263 2.2326288 2.7588182 2.6806546 4.1233120 2.8611602
82  2.6292118 3.7106134 2.0853388 2.4816060 2.2790209 2.6560547 3.3359041
83  2.0919032 3.4447249 1.9716092 2.7038636 2.5192155 1.9707250 2.7900910
84  2.9065258 1.4946050 2.1010539 2.9693934 3.1848880 3.5780768 2.8458862
85  2.5687617 3.9919162 2.0796253 1.8958550 1.3339779 3.0844118 3.1375553
86  2.4201616 1.6241933 1.3910309 2.9995983 2.9191973 3.2741134 2.2705926
87  3.2638345 2.9863761 2.0887492 3.4153172 3.1343689 3.9726100 3.3535848
88  3.7508541 3.3739402 2.6621555 4.0999327 3.8690212 4.4887840 3.6677504
89  3.8022254 2.7847457 2.7666323 3.4894945 3.6110367 4.3931446 3.8697248
90  2.1895050 2.2699430 1.3569037 1.9331474 2.0339304 2.4756338 2.7479826
91  2.7314283 3.2203905 1.8472648 2.6310005 2.3602107 3.6399942 2.7676418
92  3.0857359 2.7723143 2.4011828 2.7244549 2.9578487 3.6126644 3.2474434
93  1.8985126 3.3921840 1.5389299 1.5635480 1.2270309 2.1097251 2.7059715
94  3.5869692 2.8055172 2.4728952 3.8769363 3.8084181 4.0003078 3.7372247
95  3.8375187 3.1347300 2.6452762 3.6568571 3.5876695 4.4179908 3.9664791
96  1.4456556 1.6914846 0.9004469 2.0496666 2.1508840 2.1825769 1.6540620
97  2.0016686 2.1120583 1.1342112 1.7501024 1.8676260 2.6452762 2.3365853
98  2.9001832 1.6600056 2.1331610 3.2507155 3.3846004 3.7361186 2.6110228
99  2.0711908 2.6282730 1.5608542 2.0231761 1.9152214 3.2203905 1.8874775
100 2.7012949 3.8889571 2.2283133 2.5798571 2.2975508 3.1576156 3.1257203
101 3.5588600 2.8429833 2.3602107 3.4562570 3.3829693 4.1401075 3.6975521
102 3.4598677 2.4491128 2.5030771 3.1402726 3.2786706 4.2231849 3.4040404
103 2.1120583 1.4505707 1.3963204 2.1391954 2.3376325 2.9130610 2.1359616
104 2.2962179 2.7679933 1.6573137 2.1166109 2.2090395 2.5205970 2.8480435
105 2.2906261 2.2886418 1.6082443 1.6215795 1.8756036 2.9497419 2.5768581
106 2.7827756 2.2949816 1.9455697 3.4076013 3.3982879 3.1840000 2.9090203
107 3.2532455 2.9598506 2.1469585 3.0147434 2.9347573 3.9782382 3.3427089
108 3.3726201 3.8592867 2.5637144 3.6809121 3.3572526 4.0686743 3.4195255
109 1.5884767 3.2921591 1.4676136 1.4528659 1.0941189 2.1699630 2.2218613
110 2.0803516 3.3222954 1.9152214 2.4830039 2.3799320 2.1484525 2.6968892
111 2.7824686 3.1979221 2.0587076 1.8523834 1.9637955 3.2371133 3.2383288
112 2.9007713 2.9653024 2.2365464 3.0330548 3.0309642 3.6376140 2.8669066
113 2.2041360 3.1402726 2.1331610 1.8604993 2.0856237 2.6178022 2.6110228
114 2.9742327 3.1840000 2.3484004 2.5143462 2.6804759 3.3202751 3.3498649
115 2.3711071 1.7318013 1.6616028 3.1082196 3.1718832 2.8569700 2.4243572
116 2.9038055 3.0598281 1.7408491 2.9250044 2.5914334 3.5599002 3.1486844
117 3.3046379 3.1217083 2.2819169 3.8638739 3.5569020 4.1523999 3.1410624
118 2.6550028 2.5855958 2.1166109 3.0866399 3.0169815 3.7862148 2.1509569
119 2.6084348 2.9729539 1.8874775 2.9472450 2.7879257 3.4230246 2.5756486
120 2.3816248 3.2526648 2.0796253 1.3196460 1.6138401 2.4516995 3.1213516
           22        23        24        25        26        27        28
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
12                                                                       
13                                                                       
14                                                                       
15                                                                       
16                                                                       
17                                                                       
18                                                                       
19                                                                       
20                                                                       
21                                                                       
22                                                                       
23  1.7628702                                                            
24  1.7003980 1.0115880                                                  
25  3.3824245 2.9778928 3.1168474                                        
26  2.2721746 2.2797471 2.1968888 1.7442071                              
27  2.6605321 2.4727022 2.4817990 0.8968765 1.3240603                    
28  1.9648980 1.0115880 1.6072884 2.6088936 2.3389645 2.0543575          
29  2.0339304 2.8025366 2.9013491 2.5115933 2.4624968 2.0097877 2.4154608
30  2.2105881 1.2961786 1.4125673 1.9309840 1.8079878 1.3963204 1.0128552
31  2.4084779 2.6807337 2.3375861 4.1189636 2.4427972 3.6072293 3.3820132
32  1.6937579 2.2404557 1.3963204 3.4178722 2.1562035 2.7064028 2.6676699
33  1.2810941 1.9384681 1.2365475 3.5977540 2.2439932 2.8606159 2.4266951
34  2.5391969 1.3155002 1.4719263 2.3878886 2.1242773 1.8989303 1.0941189
35  2.8754015 1.9478633 1.7832630 3.3485443 2.4210144 2.8280318 2.1895050
36  2.3679018 2.1869827 2.5103305 3.2391403 2.1567854 2.7126194 2.1272836
37  1.7628702 0.0000000 1.0115880 2.9778928 2.2797471 2.4727022 1.0115880
38  1.5283986 0.8076269 0.9133725 2.4636843 1.6767924 1.9286335 1.2810941
39  2.5621882 2.0340019 1.8571888 1.8756036 1.0416064 1.3826348 2.0232609
40  2.3060205 2.8658617 2.2318767 3.8381562 2.2022828 3.1848880 3.3098306
41  2.2797471 2.2104205 1.2243124 3.6171648 2.5808057 2.8999242 2.5818003
42  2.4442789 2.5798571 2.3744562 3.0262534 1.3978452 2.5192661 2.8388202
43  2.4636843 1.9384681 2.4408171 2.2163862 1.5356921 1.9247088 1.7921199
44  2.1729281 1.2971677 1.0115880 2.3457531 1.8929716 1.8857149 1.6239507
45  1.0115880 1.6488101 1.7628702 2.8588539 1.5743800 2.1875287 1.7628702
46  1.2933394 1.7938948 1.2492295 3.0696110 1.7243896 2.2875589 2.0356695
47  2.1809662 2.2460246 1.7408491 2.2022828 1.0262418 1.5348055 2.3348263
48  2.5380376 1.4125673 2.0873989 2.0543575 2.2312158 1.8008938 0.8968765
49  2.3954995 2.1331610 2.6907566 2.7944331 2.2221052 2.3402891 1.6971657
50  2.5499048 1.6765018 2.1018120 1.8452839 2.1729281 1.4528659 0.9299284
51  2.9811027 1.9286335 2.1461638 1.9268953 2.2749907 1.5827387 1.3638640
52  2.0366062 1.8011098 1.6266566 4.1969359 2.8034184 3.6291386 2.6468493
53  1.4505707 2.5229239 1.8389137 3.6047601 2.2762626 2.8136272 2.7823377
54  1.8116369 1.1444998 1.6084862 2.4397779 1.7207460 1.8582154 0.9004469
55  2.6698013 2.0711908 2.1120583 2.9554101 2.7904317 2.2797471 1.3701877
56  1.9923688 2.2012098 2.1071376 1.6084862 0.6485839 0.9458482 2.0682144
57  2.2113391 1.2294788 0.7804271 3.4822258 2.5510587 2.8728727 1.7867407
58  2.4660400 1.5348055 2.1917813 2.3396599 2.4320487 1.9432650 0.6664127
59  1.8047127 1.4783678 1.6006835 3.8450686 2.5473070 3.2416935 2.0829333
60  1.9115014 2.0270520 2.2705926 2.8985223 2.0766578 2.2054997 1.6038125
61  2.3496539 1.4396997 1.6215795 2.4727022 1.8079878 1.9238227 1.2884291
62  2.2276554 2.0691750 2.4979786 1.5864721 1.3730377 1.3939884 1.8692252
63  2.1137073 1.9268953 2.2222182 2.3900894 1.1117433 2.0825502 2.1853457
64  2.8443870 2.4827971 2.5227920 1.3978452 0.9133725 1.1117433 2.2874639
65  2.4604744 1.6616028 1.5348055 1.8116369 1.5827387 1.2607256 1.4809152
66  1.4170474 1.3963204 1.8734397 2.4248558 1.9020921 1.8149337 0.9624969
67  2.8227021 1.8287827 1.4397179 3.5848854 2.5214512 3.1282355 2.5040106
68  1.9384681 1.4500335 1.7501024 2.5353790 1.7662135 1.9003968 1.1342112
69  1.9415496 2.1300628 2.0591528 2.9269832 1.2961786 2.3810361 2.4195073
70  2.3610709 1.9610521 2.0181435 3.4547747 2.1867644 2.8839222 2.2090395
71  1.6558743 1.8929716 2.0048361 1.9619406 0.8983055 1.3240603 1.7463954
72  2.2797471 1.7633875 1.2243124 2.4450333 1.7628702 1.7550755 1.7643260
73  2.6360254 1.7633875 1.8028654 3.3514678 2.2043353 2.8869428 2.2054997
74  2.1120583 2.2326288 2.0853388 2.0564995 1.3753725 1.2243124 1.8382511
75  1.8389137 1.8452785 1.4505707 4.2262675 3.1123352 3.4607205 2.2022828
76  2.3428044 1.9478633 1.7832630 3.0594032 1.7555938 2.4789036 2.1895050
77  1.7030386 2.1166109 1.4075636 2.8566639 1.6084862 2.0915112 2.3176635
78  3.0518030 3.1207269 2.6886987 3.5279685 3.0304172 2.7337453 2.6886987
79  3.3740233 3.0395085 2.7723143 3.0802303 2.4999452 2.4919439 2.7428468
80  2.4266328 2.7879768 2.2292244 2.4175411 1.8659914 1.6215795 2.5658034
81  3.4598677 3.3583339 3.3243915 2.8850849 2.2494341 2.4359583 3.0187146
82  3.3817500 3.0414002 2.4011828 3.9473833 3.4330403 3.2312517 2.8894731
83  2.8447469 2.4656127 2.0089639 3.9326832 3.5077203 3.1862786 2.2372766
84  3.6959354 3.6327205 3.5694304 1.7196967 2.3679018 1.4456556 3.0042418
85  3.1127208 2.5406567 1.9478633 3.7009588 2.6698013 3.1113162 2.8269158
86  3.4121637 2.7588182 2.9582947 1.8382511 2.3496539 1.5573507 2.0564995
87  4.3197456 3.3047702 3.3727944 2.6679559 2.9988859 2.5795403 2.9098677
88  4.5679387 3.9101225 4.0003078 3.7637172 3.8845379 3.3970453 3.2669852
89  4.5824492 4.4377635 4.2288512 2.8669066 3.2219086 2.6084348 3.8697248
90  3.2381834 2.8808351 2.4455210 2.0853388 2.1653823 1.5573507 2.5739091
91  3.2548832 2.9007713 2.7754119 3.3447941 2.6596735 2.7397330 2.5948874
92  3.5431501 3.9015116 3.5272879 3.2512966 2.9523590 2.5636560 3.3870768
93  2.6771364 2.2124750 1.4419266 3.3222954 2.5567283 2.6333344 2.3386922
94  4.6239074 4.0492576 3.9635798 3.2309507 3.7824953 2.9212934 3.3447941
95  4.7627122 4.2616059 4.0973855 3.1389538 3.4735114 2.9047909 3.7256081
96  2.2819169 2.2765560 2.1242280 2.2914097 2.0620158 1.4549339 1.6540620
97  2.8161773 2.7233346 2.3711071 2.1786366 1.8522206 1.4500335 2.3365853
98  3.5182823 3.5498203 3.6156757 2.4040030 2.6899915 1.9462153 2.7827756
99  2.3602107 2.3528347 2.3363309 2.6627763 1.6384038 2.0107315 2.1187193
100 3.1007910 2.9433415 2.4780846 4.1939235 3.3557794 3.4245561 2.8137745
101 4.5411851 3.9727991 3.8376429 2.7914093 3.2126785 2.5872440 3.4378905
102 4.1410580 4.0593020 3.9271240 2.3890511 2.6167350 2.1469585 3.5374979
103 2.8890793 2.8313915 2.7253759 1.4396997 1.5450921 0.7963504 2.3428044
104 3.0445611 3.0679096 2.5254700 3.1484624 2.8004394 2.3791134 2.6804759
105 2.9407858 3.0199408 2.6310005 2.0768050 1.5919117 1.4396997 2.7507450
106 3.6927560 3.3793628 3.3120082 3.1525906 3.4473931 2.5798571 2.5709140
107 3.9855426 3.6886109 3.4908801 3.0918922 2.9422326 2.6178022 3.2011408
108 3.9438533 3.4233064 3.3907712 4.2952033 3.8657860 3.6936732 2.9856245
109 2.0780387 1.8443054 1.1945256 3.3379079 2.3025965 2.5758409 2.0025652
110 2.6679559 2.6098538 2.1097251 3.8671179 3.3184964 3.0535043 2.3281775
111 3.3438860 3.4331006 2.8400555 3.1521608 2.4919439 2.4846224 3.2383288
112 3.2696142 3.4496258 3.2867202 3.6363976 3.1750117 2.9013510 2.8669066
113 2.2962179 3.0080521 2.4428297 3.5895576 2.6899915 2.7093543 2.7827756
114 3.4520132 3.7527172 3.2271725 3.6087475 3.1393200 2.8524112 3.3498649
115 3.2539752 3.0731929 3.0513778 2.7118141 3.0254776 2.0947215 2.2251084
116 3.9435341 2.9497419 2.8752393 2.7499263 2.7678510 2.4919716 2.6711444
117 4.1569152 3.2605089 3.5236830 3.3845439 3.4839184 3.0763065 2.6621555
118 2.9001832 2.9346426 3.1484624 3.0993240 2.5805300 2.5144870 2.3564838
119 3.0932813 2.9382643 2.8793995 3.5374979 3.0456196 2.8329187 2.3890511
120 2.9429938 3.1555827 2.3549000 3.1769344 2.4698220 2.4649757 3.1213516
           29        30        31        32        33        34        35
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
12                                                                       
13                                                                       
14                                                                       
15                                                                       
16                                                                       
17                                                                       
18                                                                       
19                                                                       
20                                                                       
21                                                                       
22                                                                       
23                                                                       
24                                                                       
25                                                                       
26                                                                       
27                                                                       
28                                                                       
29                                                                       
30  2.3551622                                                            
31  3.9964685 3.2480172                                                  
32  2.8720832 2.2986158 1.9839617                                        
33  2.8870028 2.3023508 1.7346062 0.6485839                              
34  3.0333867 0.7804271 3.2372539 2.5499048 2.4660400                    
35  4.0074330 2.0675205 2.5567283 2.5426962 2.3605990 1.4682334          
36  3.3979490 2.4612590 2.5842543 2.9901246 2.6082888 2.2114743 1.8853069
37  2.8025366 1.2961786 2.6807337 2.2404557 1.9384681 1.3155002 1.9478633
38  2.3034247 1.0262418 2.4727022 1.7501024 1.6254837 1.4158673 2.1308509
39  2.9511707 1.3753725 2.6555983 2.1926598 2.2790456 1.3754623 1.6144595
40  3.6736664 3.0316479 1.2287658 1.3939884 1.3753725 3.0598372 2.4295444
41  3.4381362 2.1983777 2.4387391 1.0416064 1.2270309 2.1926598 2.0339304
42  3.4515625 2.5947363 1.6215795 2.1941380 2.0796253 2.5429364 1.9766992
43  2.8783163 1.8706564 2.8919144 3.0459379 2.8251345 1.8450469 2.1671955
44  2.6155386 0.8968765 2.8634773 1.8267451 1.9384681 1.3155002 2.1768329
45  2.2392820 1.9402267 2.1137073 1.9211110 1.5584237 2.1378420 2.3163875
46  2.6281122 1.8857149 1.9003968 1.0115880 0.7763049 2.0048361 1.9402267
47  2.6269564 1.6767924 2.3810361 1.4617472 1.7207099 1.9268953 2.1011873
48  2.5143093 1.0115880 3.6892028 3.1132207 2.9779197 1.2294788 2.4954685
49  2.8758423 2.1867644 3.3079557 3.3241821 2.9911338 2.0903814 2.4155768
50  2.3207069 0.8036442 3.8297581 2.9995983 2.9345014 1.1202279 2.4924232
51  2.9241967 0.9133725 3.8885747 3.0646097 3.0674121 0.8142657 2.1407105
52  3.8636864 2.7805249 1.3754623 2.0235236 1.5363754 2.6555983 2.1331610
53  2.7088906 2.5914334 2.1138264 0.7693385 0.7804271 2.8425377 2.7951051
54  2.5343794 1.2114344 2.7594724 2.4312953 2.1562035 1.1606297 1.7408491
55  3.0335440 1.5701733 3.8816626 2.9208168 2.7823377 1.2884291 2.0567098
56  1.8860300 1.5635480 2.7751498 2.0563445 2.1562035 2.0313204 2.5988819
57  3.5369928 1.7442071 2.4150262 1.8989303 1.6575372 1.4125673 1.2243124
58  2.5057048 1.2124279 3.8140208 3.1976872 2.9995983 1.2810941 2.4751262
59  3.5465419 2.4230293 1.7849789 2.2328064 1.6875278 2.2079103 1.7408491
60  2.6392919 2.0089639 2.9943662 2.7060047 2.3784453 1.9350582 2.1361351
61  3.0728181 1.2287658 2.8147009 2.4817990 2.3023508 0.7804271 1.1202279
62  1.8582154 1.7046915 3.2430119 2.8969171 2.8233787 2.1872085 3.0159824
63  2.7710754 2.0631153 2.1391954 2.5213889 2.3376325 2.2229653 2.3614823
64  2.8276340 1.7408491 3.0767097 2.7721649 2.8409839 1.8692252 2.2576088
65  2.6516241 0.6002347 3.1282355 2.2188630 2.3117123 0.8076269 1.8111283
66  1.7479272 1.3972387 3.0959995 2.4879836 2.2276554 1.7463954 2.5781259
67  4.1111412 2.2234715 2.0901268 2.1680753 2.0521089 1.8598569 1.1313788
68  2.6862234 1.3939884 2.7880516 2.4636843 2.1926598 1.2270309 1.5984134
69  3.0645248 2.3097023 1.5348055 2.0339304 1.8011098 2.3207069 1.9657428
70  3.7085547 2.3675648 2.0356695 2.5192661 2.1476739 1.9937367 1.1606297
71  1.9086730 1.5690338 2.6027315 2.1562035 2.0563445 1.9249939 2.4210144
72  2.8754015 1.1313788 2.7791854 1.6915576 1.8116369 1.1202279 1.5363754
73  3.9081281 2.1869577 2.0545330 2.5227233 2.2435265 1.7338666 0.7804271
74  2.2684224 1.5340442 3.0286045 2.1983777 2.2022828 1.7139129 2.1680753
75  3.5541822 2.5127218 2.5192155 1.9432650 1.4528659 2.3265794 2.0496666
76  3.5148140 2.0675205 1.9384681 2.1476739 1.9286335 1.7555938 0.9624969
77  2.6117140 1.8811528 2.1304350 0.7763049 1.0115880 2.1071376 2.1580908
78  3.5126911 2.5636560 3.8875605 2.7734971 2.7827756 2.3909669 2.4429920
79  3.9064282 2.4755254 3.4952803 3.0175922 3.0147434 2.0968846 1.7257719
80  2.3922469 1.9684581 3.2749288 1.8128124 2.1407105 2.3097023 2.7490858
81  3.8023080 2.8557838 3.5182823 3.4561499 3.3896816 2.6292118 2.3466228
82  4.1045666 2.6239919 3.8795397 2.6659077 2.7499263 2.3281775 2.2392820
83  3.5874288 2.2705926 3.9457781 2.6112157 2.5429364 2.0796253 2.4482883
84  2.9123966 2.5133541 4.5655338 3.6613152 3.8342641 2.7312560 3.4224214
85  4.2732411 2.4954685 2.5401358 2.2433964 2.2318767 2.0564995 1.0115880
86  3.1147014 1.7897806 4.2580441 3.5867755 3.5891703 1.6914846 2.5299401
87  4.3904134 2.5030771 4.5044251 4.0811518 4.1282068 2.0595794 2.3376325
88  4.7363333 3.3003205 5.1367463 4.6413024 4.5516536 2.8137745 2.9065258
89  4.1515560 3.3851585 5.0614064 4.2566702 4.4518614 3.3447941 3.6242045
90  3.0773540 1.7633875 3.8205694 2.5758409 2.8870028 1.9086730 2.5200230
91  3.9729020 2.5657858 3.4245561 3.1566390 3.0175922 2.1009783 1.5884767
92  3.4467573 3.0936584 4.3155873 3.2835876 3.4168441 3.1244634 3.2771958
93  3.5769339 1.9070010 2.9134834 1.7783927 1.8929716 1.6826343 1.5538377
94  4.3231481 3.0473071 5.4354771 4.4552513 4.5543540 2.8181406 3.3660442
95  4.5999937 3.2867202 5.0979294 4.3949145 4.5362161 3.0276447 3.1997113
96  2.2453858 1.4419266 3.5841655 2.4266442 2.4372436 1.6241933 2.3833222
97  2.8429833 1.7857650 3.4677160 2.4237093 2.5947781 1.8522206 2.2431709
98  2.9717083 2.6621555 4.5999627 3.8295192 3.8362444 2.7340354 3.3204286
99  3.0331017 2.0829756 2.7088906 2.5497595 2.3754509 1.9313764 1.7628702
100 4.1841487 2.8524112 3.4495280 2.7418915 2.5872440 2.4375702 1.8851104
101 4.3337962 2.9565967 4.9239117 4.1918128 4.3397325 2.7244549 3.0213156
102 3.7307087 3.0718503 4.5575108 3.9353391 4.0925392 3.0833513 3.3485443
103 2.4375743 1.7938948 3.6940868 2.8211328 2.9693934 2.0591528 2.7135936
104 3.2656125 2.2985613 3.8840309 2.5133541 2.6851118 2.2765560 2.5574565
105 2.8873380 2.1187193 3.3359430 2.4175411 2.6657997 2.2790209 2.5281523
106 3.4556726 2.5143462 4.9454044 3.7971206 3.8084181 2.3995958 3.1143719
107 4.1216289 2.9007713 4.3130226 3.7067046 3.7630201 2.6084348 2.5524547
108 4.6890459 3.2214399 4.4269654 4.0126617 3.7994730 2.6391364 2.2906261
109 3.2857235 1.8499291 2.4371990 1.5573507 1.4158673 1.6254837 1.3240603
110 3.4276720 2.3602107 3.7785554 2.4474404 2.3744562 2.2104205 2.4333226
111 3.7072727 2.7219748 3.4841760 2.5282942 2.7666323 2.6489125 2.4612590
112 3.6203418 2.9676899 4.1667127 3.4280842 3.3108901 2.7676418 2.7736530
113 2.9717083 2.6621555 3.2405427 2.0016686 2.0145052 2.7340354 2.7335689
114 3.6718172 3.0528721 4.1350411 2.9653024 3.1122170 3.0080521 3.0228969
115 2.8483845 2.1595012 4.6611178 3.4857309 3.4980343 2.2090395 3.0749443
116 4.2359665 2.2210532 4.0364456 3.5579274 3.6118055 1.7057403 1.8011098
117 4.4188342 2.7827756 4.7032626 4.3620623 4.2149264 2.2283133 2.4266442
118 3.2528199 2.6489125 3.7386483 3.5119402 3.2662310 2.4680109 2.5553013
119 3.6666794 2.5948874 3.9042358 3.3047702 3.1135017 2.2365464 2.1895050
120 3.3244882 2.4954685 3.2678159 1.7832630 2.2090395 2.6281122 2.6840007
           36        37        38        39        40        41        42
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36                                                                       
37  2.1869827                                                            
38  2.3204408 0.8076269                                                  
39  2.0435491 2.0340019 1.6364443                                        
40  2.6305679 2.8658617 2.5030771 2.3023508                              
41  3.0645742 2.2104205 1.9635620 2.1562035 1.8221893                    
42  1.6462905 2.5798571 2.2684224 1.5507228 1.4125673 2.4882547          
43  1.2243124 1.9384681 1.8756036 1.5864721 2.9995571 3.2191119 1.8571888
44  2.8665171 1.2971677 0.8076269 1.5884767 2.7340354 1.7897806 2.6063654
45  1.4783678 1.6488101 1.4075636 1.8877542 2.0557823 2.3878886 1.6336280
46  2.0265761 1.7938948 1.4500335 1.6777891 1.3638640 1.3383450 1.5864721
47  2.4333226 2.2460246 1.6384038 0.9458482 1.7442071 1.7042775 1.5363754
48  2.4243572 1.4125673 1.4865267 1.9794547 3.7096302 3.0916760 3.0147728
49  1.0115880 2.1331610 2.2622022 2.1861400 3.3079557 3.4377381 2.3225800
50  2.5100126 1.6765018 1.6084862 1.8287827 3.6480571 2.9234945 3.0122525
51  2.6205314 1.9286335 1.8857149 1.5984134 3.6232738 2.7744504 2.9821900
52  2.4201616 1.8011098 1.9761763 2.7218349 2.0089639 2.1378209 2.2705926
53  2.8445738 2.5229239 2.0853388 2.4391040 1.3569037 1.5119983 2.1707661
54  1.3196460 1.1444998 1.2243124 1.4809152 2.7153582 2.4636843 2.0097877
55  2.4060143 2.0711908 2.2210532 2.1287737 3.4359942 2.4905300 3.0574211
56  2.2749709 2.2012098 1.5573507 1.2270309 2.3892585 2.4979786 1.8028654
57  2.3605990 1.2294788 1.4911750 1.9716092 2.3644944 1.3826348 2.4103437
58  2.2292244 1.5348055 1.7243896 2.1301856 3.7501626 3.1264091 3.0645094
59  1.6254837 1.4783678 1.8028654 2.3918437 2.1670291 2.2680317 2.0097877
60  1.0379931 2.0270520 2.0587076 1.9587948 2.7306343 2.7880516 2.0303792
61  1.4705011 1.4396997 1.5236535 1.1213722 2.6545903 2.2526389 1.9268953
62  2.3204408 2.0691750 1.6440267 1.8985126 3.2662310 3.3222954 2.4486247
63  1.6384038 1.9268953 1.6144595 1.6241933 2.4175411 2.9471738 1.4396997
64  2.0948983 2.4827971 2.0595794 0.8220133 2.7775457 2.9073622 1.7442071
65  2.4228687 1.6616028 1.3155002 0.9004469 2.7915868 2.0385034 2.3097023
66  1.8970860 1.3963204 1.2810941 2.0232609 3.0169815 2.7553753 2.4912044
67  2.5553013 1.8287827 1.9202407 1.9331474 2.2366949 1.7338666 2.1872085
68  1.1250933 1.4500335 1.5138224 1.4263823 2.6263168 2.4312953 1.8877542
69  1.3638640 2.1300628 1.8522206 1.5645881 1.5348055 2.4147733 0.6002347
70  0.9367199 1.9610521 2.1229700 1.8384526 2.0941509 2.4230293 1.3972387
71  1.7429337 1.8929716 1.4158673 1.3754623 2.3784453 2.5808057 1.6616028
72  2.4163543 1.7633875 1.4419266 1.0470754 2.2576088 1.3328253 2.1011873
73  1.5283986 1.7633875 1.9571754 1.6875278 2.2576088 2.2976486 1.6320556
74  1.9247088 2.2326288 1.8523834 1.1899660 2.4312953 2.2986158 1.8582154
75  2.4817990 1.8452785 2.1120583 2.7704307 2.3389645 1.6405289 2.7324167
76  1.3044468 1.9478633 1.9010853 1.2961786 1.7673383 2.0339304 1.0621387
77  2.4989369 2.1166109 1.5884767 1.5584237 1.3826348 1.1706862 1.7338666
78  2.8758423 3.1207269 3.0129155 2.4297606 3.0122525 2.3106264 2.9714814
79  2.2650718 3.0395085 2.9404208 1.7233786 2.8158313 2.5985710 2.2433964
80  3.0219665 2.7879768 2.2226875 1.7338666 2.3900894 1.9331474 2.4303107
81  1.8851104 3.3583339 3.1933235 1.9060594 2.9943151 3.3829693 1.9635620
82  3.4338038 3.0414002 3.0185697 2.6401439 3.1057270 1.7938948 3.3433517
83  3.3401848 2.4656127 2.5976617 2.8500649 3.3811573 1.8877542 3.6006506
84  3.2593163 3.6327205 3.2193452 2.1941380 3.8983706 3.6795260 3.1720508
85  2.5905024 2.5406567 2.5218978 1.8857149 2.0756326 1.5450921 2.1277156
86  2.5312700 2.7588182 2.6427583 1.7867407 3.8561238 3.3856946 2.9763420
87  3.0702635 3.3047702 3.2942212 2.1201411 4.1557327 3.6171648 3.2895577
88  3.1605421 3.9101225 4.0533781 3.1048437 4.6329160 4.1253496 3.8749175
89  3.7738026 4.4377635 4.1135135 2.7767456 4.2507068 4.0466360 3.5962648
90  3.2593354 2.8808351 2.4473782 1.5363754 3.0958229 2.2875589 2.8287061
91  1.8522206 2.9007713 2.9433415 1.9478633 2.8836084 2.7588182 2.2318767
92  3.2587815 3.9015116 3.5767016 2.6405389 3.3279073 3.1566390 3.0809781
93  2.8584161 2.2124750 2.0803516 1.8116369 2.2862933 0.9133725 2.5143093
94  3.8719346 4.0492576 3.9774728 3.0289897 4.7570220 3.9551450 4.1825675
95  3.6994912 4.2616059 4.0930160 2.7462888 4.3949117 3.9646556 3.7070225
96  2.3884455 2.2765560 2.0145052 1.6881583 2.9511527 2.3265794 2.5862938
97  2.5826668 2.7233346 2.3420338 1.3082827 2.6933140 2.2542514 2.2876962
98  2.8077872 3.5498203 3.3186665 2.4751262 3.9691774 3.8054978 3.1907892
99  1.1117433 2.3528347 2.2041360 1.3753725 2.2790456 2.5758409 1.3638640
100 2.6940715 2.9433415 3.0562845 2.6561557 2.7588340 2.0587076 2.8754015
101 3.5463264 3.9727991 3.7913886 2.4728952 4.2667839 3.7804236 3.5541822
102 3.2391403 4.0593020 3.6986250 2.3281775 3.8381562 3.8712276 3.0262534
103 2.5930928 2.8313915 2.3711071 1.3826348 3.0837214 2.8999242 2.3900894
104 3.1761048 3.0679096 2.8123782 2.2090395 2.9672743 2.0689960 2.9997665
105 2.7289666 3.0199408 2.5143462 1.3196460 2.5213889 2.4450333 2.0825502
106 3.3534522 3.3793628 3.3447941 2.8276340 4.2633374 3.3846004 3.8492962
107 2.8665171 3.6886109 3.5461905 2.2494341 3.5782984 3.3270528 2.9273817
108 2.7054993 3.4233064 3.7015084 3.0813506 3.9435341 3.4026864 3.4953594
109 2.2079103 1.8443054 1.7923174 1.7243896 1.8853069 1.0115880 2.0521089
110 3.1135474 2.6098538 2.6489125 2.7277280 3.0832397 1.8287827 3.3224731
111 3.0645248 3.4331006 3.0742925 1.9637955 2.4698220 2.2283133 2.4199292
112 2.5441698 3.4496258 3.4083374 2.7221904 3.4005221 3.1635080 3.0175020
113 2.8077872 3.0080521 2.7314283 2.4751262 2.2573943 1.9553192 2.5745464
114 3.3608575 3.7527172 3.4809608 2.6804759 3.0901841 2.6550028 3.1213972
115 3.1003244 3.0731929 2.9578487 2.5510775 4.0098896 3.2297372 3.5665460
116 2.8450308 2.9497419 2.9379183 1.7783927 3.6432351 3.0144533 2.9326883
117 2.6442183 3.2605089 3.4942209 2.7059715 4.3980070 3.8908331 3.5292627
118 1.4682334 2.9346426 2.9565967 2.3557018 3.3465323 3.4857309 2.4675759
119 2.1272836 2.9382643 3.0415932 2.4486949 3.3098306 2.9271513 2.8388202
120 3.4443336 3.1555827 2.6804759 2.0930607 2.2431709 1.5450921 2.6461522
           43        44        45        46        47        48        49
2                                                                        
3                                                                        
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42                                                                       
43                                                                       
44  2.3471348                                                            
45  1.5584237 2.0979082                                                  
46  2.2430309 1.7938948 1.1202279                                        
47  2.1707661 1.5984461 1.7554141 1.2124279                              
48  1.6241933 1.6740068 2.1711566 2.5596840 2.4827971                    
49  0.9825966 2.8134791 1.6084862 2.3831905 2.6409354 1.8522206          
50  1.8047127 1.6558743 2.2124781 2.4450333 2.2699430 0.6143829 1.9319985
51  2.0136302 1.6863972 2.5611432 2.5414208 2.1968888 1.1342112 2.2405809
52  2.7775457 2.4230293 1.9846402 1.8111283 2.6366486 3.1093216 2.9995571
53  3.0767097 2.3429583 1.7356257 0.9458482 1.6777891 3.3222954 3.1803384
54  1.0470754 1.7298913 1.1706862 1.6084862 1.8776540 1.2364476 1.1202279
55  2.4371990 2.2423303 2.4210144 2.2461619 2.4320487 1.9060594 2.1580908
56  1.6875278 1.7783927 1.4583381 1.6084862 1.0178348 2.0465962 2.1494648
57  2.5100351 1.5450921 2.0860119 1.5690338 2.0903814 2.2985621 2.6955097
58  1.6636069 1.9923688 2.1071376 2.5227920 2.6045597 0.6002347 1.5608542
59  2.1562035 2.3705692 1.4987320 1.6006835 2.4804554 2.6140070 2.1926598
60  1.4505707 2.5687617 1.2294788 1.7207099 2.1693666 2.0803516 0.9064062
61  1.3221091 1.7170984 1.6996436 1.7093117 1.7473266 1.5138224 1.5480251
62  1.3196460 2.0691750 1.6826343 2.3781177 2.1120583 1.5119983 1.8278773
63  1.0178348 2.1580908 1.2761367 1.9286335 1.8659914 2.0903814 1.7679150
64  1.3569037 2.1331610 2.0496666 2.1917813 1.4125673 2.0417971 2.0257103
65  1.8821664 1.0384025 2.0521089 1.8116369 1.3155941 1.4505707 2.3054282
66  1.5117187 1.8877542 1.1202279 1.7937530 2.1272079 1.3155941 1.3978452
67  2.6245583 1.8659914 2.4868200 1.9813098 2.1580908 2.7484653 3.0769640
68  1.1202279 1.9455697 1.2365475 1.5584237 1.8349499 1.5236535 1.0470754
69  1.5389299 2.3412813 1.0883102 1.3240603 1.5503690 2.6401439 1.9554199
70  1.6914846 2.5441698 1.6241933 1.6746806 2.1679165 2.5916719 1.7867407
71  1.2455510 1.8929716 0.9299284 1.4719263 1.3638640 1.8342352 1.5864721
72  2.2435265 1.1945256 1.9813098 1.3383450 1.0621387 2.0564995 2.5473070
73  1.8116369 2.2202831 1.9813098 1.8821664 2.1519536 2.4455210 2.1765547
74  1.7460199 2.0270520 1.5450921 1.4676136 1.1872666 2.0803516 1.8384526
75  3.0358710 2.4278900 2.0545330 1.5480251 2.6088936 2.9921975 2.8724583
76  1.6863972 2.1768329 1.6082443 1.3826348 1.6006835 2.4954685 1.9955477
77  2.5154602 1.6725422 1.5864721 0.6485839 0.8036442 2.7159013 2.7897853
78  3.2203905 2.9655532 2.8316592 2.2326288 2.3447095 3.2281179 2.9906560
79  2.5320985 2.9037319 2.7418915 2.3326503 2.0803516 3.0032797 2.5687617
80  2.8434850 2.0829756 2.2749709 1.6657411 1.1117433 2.8313915 3.0370515
81  2.0938596 3.3685524 2.6035674 2.6371024 2.3217555 3.1178368 2.1380510
82  3.7070225 2.7383813 3.3079975 2.4511598 2.5695280 3.4217463 3.6659465
83  3.5561843 2.4375743 3.0067135 2.3551622 2.7847773 2.9218963 3.3710330
84  2.7469632 3.1245210 3.1567646 3.1483188 2.3892585 2.8480435 2.8925758
85  2.9213217 2.3909669 2.7188678 1.9350582 2.0095701 3.1979221 3.1878301
86  1.9937367 2.5952623 2.7915868 2.8999242 2.4266951 1.8443054 2.0751646
87  2.6942982 3.0508361 3.6398369 3.5003459 2.9250044 2.6738992 2.9811715
88  3.3067661 4.0119834 3.9640185 3.8699638 3.7014467 3.3937131 3.0316207
89  3.5585524 3.9310005 3.9936609 3.7520846 3.0112840 3.8220986 3.6524205
90  2.8783163 2.0145052 2.9122311 2.3745791 1.5507228 2.6012979 3.1984265
91  2.3564838 3.0479773 2.5872440 2.3363309 2.3642855 2.9382643 2.2210532
92  3.4095077 3.5489038 3.1761258 2.7767456 2.4678608 3.6999098 3.2692074
93  2.9812030 1.7923174 2.5429364 1.6320556 1.6996436 2.7912585 3.1663617
94  3.6859688 3.7107160 4.1779236 3.9138595 3.4528096 3.3632099 3.6008926
95  3.5420037 3.8465564 4.1468143 3.8517951 3.2037265 3.6761153 3.6362990
96  2.2392820 2.0587076 1.9937367 1.8011098 1.6937579 2.0219033 2.1304350
97  2.4097928 2.2041360 2.3610709 1.9286335 1.3155002 2.5143462 2.5745464
98  2.5855958 3.4142047 2.9555971 3.1000023 2.7310090 2.8202416 2.3612499
99  1.4456556 2.5320985 1.5180038 1.6215795 1.6616028 2.3989069 1.4505707
100 3.3350217 3.0773709 2.8865580 2.1960953 2.6645391 3.4636950 3.1137564
101 3.2866110 3.5238962 3.9309147 3.6618894 2.9727045 3.3370153 3.4344984
102 2.9525263 3.6211385 3.4609376 3.3652687 2.5985710 3.4395453 3.1163084
103 2.0829493 2.3363309 2.3279725 2.2875589 1.5348055 2.2684224 2.3402891
104 3.2730095 2.6048023 2.8854117 2.1835071 2.0089639 3.1074157 3.2264128
105 2.4861045 2.3989069 2.4266328 2.0231761 1.1389480 2.8329187 2.7944331
106 3.3016845 3.2259764 3.4239195 3.2071020 3.0645094 2.8480185 2.9973928
107 2.9674921 3.4529995 3.3542560 3.0508361 2.6207153 3.3425526 2.9454943
108 3.2989617 3.7694328 3.4847398 3.2359678 3.4952534 3.4551338 2.8865580
109 2.5363790 1.8443054 1.8929716 1.0416064 1.5984134 2.5952623 2.6008730
110 3.4377265 2.5833814 2.7836384 2.0979082 2.5708781 3.0445967 3.1966065
111 3.2064511 2.9018772 2.9454943 2.2124750 1.7283572 3.5310039 3.3578707
112 3.0266367 3.5550150 2.8557838 2.6452762 2.8048358 3.3530806 2.5709140
113 3.1997113 2.8467432 2.2765560 1.5825046 1.9762499 3.3921394 3.0213156
114 3.6316195 3.3846436 3.2153491 2.5709140 2.4244239 3.7850826 3.5002335
115 2.9382505 2.9036779 2.9987030 2.8870028 2.7277280 2.4770206 2.6521022
116 2.6105817 2.6621555 3.3208257 3.0238520 2.5169667 2.5895233 2.9057318
117 2.6659077 3.5090686 3.5105057 3.5579274 3.4381362 2.7012949 2.4489303
118 1.8874775 3.3319250 2.1786366 2.5396298 2.7031124 2.6560547 1.2828929
119 2.6489125 3.2011408 2.6292118 2.4589575 2.7118141 2.9007713 2.1869827
120 3.4547747 2.3909669 2.8488255 1.9086730 1.4783678 3.4609376 3.6828805
           50        51        52        53        54        55        56
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51  0.7693385                                                            
52  3.3229200 3.4195618                                                  
53  3.1663870 3.3120746 2.2501867                                        
54  1.3076471 1.5645881 2.2875589 2.4817990                              
55  1.5180038 1.3730377 3.2649598 2.9441553 1.6241933                    
56  1.8692252 2.0860119 2.9641863 2.0873989 1.6152537 2.4810117          
57  2.3376325 2.2074066 1.5507228 2.2768272 1.7042775 2.0231761 2.5611432
58  0.6485839 1.1899660 3.1803384 3.2939556 1.1202279 1.5236535 2.1729281
59  2.8088898 2.9373214 0.8968765 2.2876962 1.6285315 2.6535828 2.6481205
60  1.9684581 2.2012098 2.7088906 2.4636843 1.0332048 1.7204919 1.8860300
61  1.4682334 1.3076471 2.3954995 2.6376211 0.7693385 1.4397179 1.8221893
62  1.6320556 2.1351108 3.1876234 2.9582947 1.5573507 2.7770989 1.2243124
63  2.3034247 2.5456423 2.3117123 2.6082888 1.4528659 2.9863761 1.4590577
64  1.8960856 1.8079878 3.2785250 2.8919144 1.6937579 2.4766477 1.1202279
65  1.1872666 0.9367199 2.8445738 2.5380376 1.3835622 1.6375035 1.4437454
66  1.3383450 1.9268953 2.6468493 2.4266442 0.9004469 1.9313764 1.5571480
67  2.8588539 2.6027315 1.5984134 2.6618620 2.1765547 2.7583897 2.7665988
68  1.4783678 1.6144595 2.3892585 2.4496495 0.3981752 1.4705011 1.6636069
69  2.6967122 2.7951051 1.9384681 1.9870751 1.5984134 2.8450308 1.6040434
70  2.6719691 2.6152942 1.7897806 2.5227233 1.4682334 2.3833222 2.3893899
71  1.7628702 2.0901268 2.6555983 2.0915112 1.1606297 2.3163875 0.6485839
72  1.7937530 1.5389299 2.5192661 2.0155799 1.5864721 1.6278492 1.6392224
73  2.5923572 2.4230293 1.6863972 2.7505142 1.5864721 2.4804554 2.4879341
74  1.7318013 1.7783927 3.0262534 2.0945396 1.3939884 1.7003980 1.0332048
75  2.9693934 3.0296008 1.5963766 1.9408877 2.1421565 2.2940766 2.9885167
76  2.4924232 2.3471348 1.9036743 2.2435265 1.4505707 2.2707830 1.9937367
77  2.5227920 2.5380376 2.2392820 0.9367199 1.9619406 2.4155768 1.4719263
78  2.7754119 2.5282942 3.6592893 2.6111647 2.5709140 1.5884767 2.7418915
79  2.6790944 2.2494341 3.4796784 3.0064448 2.2849645 1.9217330 2.5033798
80  2.4108835 2.3605990 3.3937997 1.8011098 2.3705692 2.2846973 1.4783678
81  2.8838533 2.6693477 3.7579633 3.3092928 2.3337193 2.5781259 2.3484004
82  3.0508361 2.6427583 3.4590700 2.8071199 2.9273817 1.9152214 3.2440380
83  2.6020963 2.4266442 3.1992847 2.7236973 2.5820616 1.4922957 3.1992446
84  2.3642855 2.2433964 4.6854992 3.6119039 2.7829181 2.5905024 2.0761999
85  3.0751558 2.6657997 2.3642855 2.5429364 2.4383106 2.4573297 2.8260123
86  1.4396997 1.1444998 4.0328530 3.6531617 1.9286335 1.6777891 2.1671955
87  2.4375743 1.7687489 4.3012790 4.3072843 2.7059715 2.3692405 3.0682206
88  3.0590945 2.6084348 4.8091289 4.6126886 3.1284680 2.2283133 3.7815164
89  3.3370153 2.9598506 5.2497097 4.2041976 3.5780439 3.0566357 3.0802303
90  2.1359616 1.7298913 3.7872972 2.8147009 2.4868019 2.0820909 1.9483337
91  2.6886987 2.3484004 3.2881316 3.0811914 2.0968846 1.8278773 2.6693477
92  3.1963364 3.0185697 4.4585206 3.0112840 3.1054258 2.5107251 2.6488856
93  2.5421244 2.1983777 2.5498827 2.1236735 2.2404557 1.8952749 2.4730950
94  2.8557838 2.3484004 5.1745372 4.5061109 3.3954256 2.2622022 3.5506960
95  3.2371133 2.6790944 5.1066331 4.4494618 3.4923256 2.7856512 3.4073818
96  1.5236535 1.5573507 3.3314642 2.3457531 1.6138401 1.1606297 1.6082443
97  2.0587076 1.7938948 3.4940604 2.4450333 2.0504023 1.7662135 1.6038125
98  2.3564838 2.3421920 4.5795898 3.6306747 2.5524547 2.2171286 2.3602107
99  2.2326288 2.1979739 2.7642250 2.3954995 1.3639023 1.9813098 1.6540620
100 3.1866975 2.8950303 3.0414002 2.6659077 2.6239919 1.9036743 3.2359678
101 2.9013510 2.3310016 4.9006771 4.2861193 3.2330110 2.5787688 3.1410624
102 3.0187146 2.7418915 4.8292687 3.8596395 3.1237093 2.9554101 2.5282942
103 1.8382511 1.7717889 3.7998677 2.8136272 2.0216672 2.1361351 1.2364476
104 2.6110228 2.3281775 3.7219395 2.4826210 2.6300011 1.7298913 2.4784334
105 2.4375743 2.2318767 3.6118240 2.4266442 2.3333325 2.3853046 1.4419266
106 2.2962179 2.0541032 4.5479508 3.7310629 2.7589320 1.3978452 3.0623092
107 2.9284096 2.4780846 4.3370990 3.6545067 2.8400555 2.2423303 2.8698557
108 3.1963364 2.8400555 3.9411455 3.9285094 2.8273139 1.9089230 3.7668160
109 2.4175411 2.2542514 2.0181435 1.7046915 1.7501024 1.7418091 2.2171286
110 2.6806546 2.5396298 3.1917434 2.4103437 2.5103305 1.4676136 2.9905656
111 3.1225238 2.7734971 3.7099883 2.5030771 2.8624539 2.4988259 2.3989069
112 2.9404208 2.8080285 4.0094016 3.1135017 2.6048023 1.8609395 2.9068211
113 3.0175922 3.0064448 3.2115577 1.5884767 2.5524547 2.2171286 2.3602107
114 3.2945536 3.0473071 4.1744792 2.7462888 3.1394546 2.3642855 2.8557838
115 1.9010853 1.8278773 4.3083831 3.3921840 2.4097928 1.3076471 2.5781259
116 2.3447095 1.6384038 3.8084181 3.8151992 2.4474404 2.0690537 2.8428261
117 2.4919439 2.0711908 4.2694336 4.4041990 2.5293319 1.9286335 3.4330403
118 2.4428297 2.5107251 3.6032658 3.2321555 1.8477391 2.0095701 2.4297606
119 2.5636560 2.3801614 3.5655822 3.1054669 2.1484525 1.3701877 2.8429833
120 3.0751558 2.8342351 3.4159885 1.9202407 2.9191992 2.6391039 2.2819169
           57        58        59        60        61        62        63
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53                                                                       
54                                                                       
55                                                                       
56                                                                       
57                                                                       
58  2.3265794                                                            
59  1.4155670 2.5610818                                                  
60  2.3025965 1.7367557 1.9286335                                        
61  1.4719263 1.4505707 1.8011098 1.3754623                              
62  2.9250044 1.7463954 2.7802106 2.0587076 2.0431155                    
63  2.4619564 2.2846973 1.9716092 1.9217330 1.7717889 1.2961786          
64  2.6955097 2.1882379 2.8588539 2.0123079 1.5480251 1.5701733 1.5450921
65  1.7565458 1.6239507 2.5135119 2.0620158 1.1117433 1.8929716 2.0526203
66  2.2461619 1.1706862 2.0829333 1.2828929 1.6082443 1.2810941 1.7096594
67  1.0128552 2.9138631 1.7207460 2.8181078 1.7867407 3.1653392 2.4170864
68  1.7501730 1.3155002 1.6765018 0.7693385 0.6582843 1.7938948 1.6082443
69  2.1746061 2.6807337 1.5984134 1.6937579 1.7173141 2.0689960 1.0384025
70  1.6971657 2.4924232 1.1313788 1.5356921 1.3076471 2.6851118 1.7938948
71  2.3876657 1.8692252 2.2079103 1.3753725 1.5690338 1.0384025 1.1117433
72  1.3826348 2.1083557 2.2680317 2.0543575 1.2335054 2.3892209 2.2656095
73  1.3826348 2.4892864 1.2810941 2.0543575 1.2335054 2.7312560 1.8389137
74  2.3174748 1.9553192 2.5192661 1.2971677 1.4002269 1.8523834 1.9571754
75  1.2810941 2.8211328 1.3826348 2.2543062 2.1917813 3.4121637 2.9580557
76  1.5573507 2.4751262 1.4505707 1.6462905 1.1202279 2.5133541 1.6725422
77  1.8221893 2.7553753 2.1679165 2.1508840 1.9442874 2.4649757 2.1476739
78  2.6098538 2.9259901 3.1799944 2.2292244 2.2699880 3.5646717 3.5420037
79  2.5282942 2.8226432 2.9578487 2.0947215 1.7196967 3.2401968 2.9440886
80  2.6205314 2.8134791 3.1619702 2.4005240 2.2876962 2.5916719 2.7523933
81  3.2010433 2.9442371 3.1424169 1.9200627 2.0270520 2.9018772 2.5694735
82  2.1768329 3.2510897 3.2087235 2.9821900 2.4698220 4.0460118 3.9031419
83  1.9086730 2.6881101 2.8971339 2.7298957 2.3745791 3.7424984 3.7830842
84  3.7843744 2.7736530 4.1776023 2.6932093 2.6535828 2.6098538 3.1650894
85  1.4946050 3.1956192 2.2460246 2.6920584 1.8571888 3.5625856 2.9050614
86  2.9744174 1.7020202 3.4170121 2.0941509 1.6765018 2.2987225 2.7312560
87  3.1289072 2.6711444 3.7951199 3.0145152 2.0887492 3.3173254 3.3437331
88  3.6695395 3.0562845 4.1070629 2.9050614 2.6621555 4.0721774 4.1252515
89  4.2484559 3.6935691 4.7444837 3.3578707 3.2064511 3.7808130 4.0182414
90  2.6366486 2.6587805 3.5127004 2.7671243 2.1138264 2.8004394 3.0308583
91  2.4436879 2.6790944 2.6550028 1.7777076 1.5927007 3.2428475 2.8657681
92  3.6154053 3.4520132 3.9747390 2.6165653 2.8970249 3.5767016 3.6979779
93  1.3082827 2.7730868 2.4228687 2.5597186 1.8011098 3.2649519 2.9798023
94  3.8475507 3.1393200 4.6083232 3.3494230 2.9567356 3.9966291 4.3530220
95  3.9484829 3.5422928 4.5856705 3.4052366 2.9578487 3.9973909 4.1176643
96  2.3471348 1.7832630 2.8077872 1.5503690 1.6006835 2.2326288 2.5812807
97  2.5103305 2.4297606 3.0795031 2.0435491 1.7429337 2.5320985 2.6392919
98  3.7281140 2.5320985 3.9175325 2.1407105 2.5103305 2.7314283 3.2064511
99  2.2995033 2.2210532 2.1242280 0.9825966 1.2492295 2.2041360 1.7857650
100 2.0938596 3.1840000 2.6192161 2.3892209 2.2283133 3.9589042 3.5877896
101 3.7210938 3.2383288 4.3914002 3.2381834 2.7059153 3.6879509 3.8592867
102 3.9913581 3.3438860 4.3115653 2.8985223 2.8313915 3.1822917 3.3793628
103 2.9812030 2.2460246 3.3380759 2.0567098 1.9238227 1.9610521 2.3675648
104 2.5913640 2.9184360 3.3608575 2.5128467 2.3181378 3.3968655 3.4881089
105 2.8398898 2.8303187 3.2746954 2.3144459 2.0827464 2.5143462 2.5441698
106 3.2929268 2.4830410 3.9424624 2.5817526 2.5322551 3.4914789 3.9245394
107 3.3186665 3.1310057 3.7743657 2.5687617 2.3420338 3.5569431 3.5024909
108 2.9193392 3.0590945 3.2627931 2.4984589 2.3868263 4.1721417 3.9488225
109 1.0414878 2.4961564 1.7429337 1.9309840 1.4676136 2.9356514 2.5229239
110 2.0504023 2.7642250 2.8333905 2.4473809 2.3279725 3.6535989 3.6520035
111 2.8480435 3.4378905 3.4200012 2.7019833 2.4473782 3.4862728 3.2836832
112 3.1769534 2.9433415 3.3449204 1.9089230 2.4244239 3.5416330 3.5328431
113 2.6063654 3.1566390 2.8707782 2.1407105 2.5103305 3.3186665 3.2064511
114 3.2528199 3.5431501 3.7728812 2.7426484 2.8534293 3.8496639 3.8488890
115 3.1567646 2.1229700 3.7065355 2.2431709 2.3212062 2.9707316 3.5148140
116 2.5848249 2.5866787 3.3606836 2.7847773 1.7408491 3.2496987 3.1381895
117 3.1742286 2.3989069 3.5499974 2.5135507 2.0812144 3.5160111 3.5434289
118 3.0700166 2.2602898 2.8004782 1.0384025 1.8985126 2.6391364 2.5833814
119 2.6452762 2.4802562 2.8537089 1.6038125 1.8874775 3.3322800 3.2377926
120 2.5812807 3.4588097 3.3352298 2.9828080 2.6366486 3.4152889 3.3036842
           64        65        66        67        68        69        70
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64                                                                       
65  1.3963204                                                            
66  2.0751122 1.7662135                                                  
67  2.6675060 2.0445213 2.8500649                                        
68  1.6462905 1.4397179 1.1342112 2.2126757                              
69  1.7565458 2.1580908 2.0003037 2.1236735 1.5480251                    
70  2.2234715 2.2392820 2.2090395 1.7408491 1.2961786 1.2270309          
71  1.2810941 1.5827387 1.0941189 2.6766899 1.2270309 1.2961786 1.9937367
72  1.7673383 0.7763049 2.0097877 1.7338666 1.5356921 2.0136302 2.0235236
73  2.2079103 2.0261824 2.4063727 1.1202279 1.5356921 1.5176705 0.7693385
74  1.2761367 1.3275212 1.5661312 2.6967122 1.2114344 1.7338666 2.0441311
75  3.3880418 2.6166450 2.4034246 2.0690537 2.1048256 2.4280425 1.9619406
76  1.8011098 1.8111283 2.1895050 1.4854017 1.2761367 1.0416064 0.6485839
77  2.1018120 1.6915576 2.1083557 2.1011873 1.9211110 1.6266566 2.1304350
78  2.8161773 2.3310016 2.8557838 3.1555827 2.2765560 2.9884469 2.7080069
79  2.0270520 2.0451185 2.9068211 2.6550028 1.9478633 2.4373949 2.0675205
80  2.0265761 1.6971657 2.3784336 2.9727917 2.2680317 2.4214077 2.8834981
81  1.7196967 2.4919439 2.8611602 3.2350582 2.0048327 2.1825769 2.1242280
82  3.2881742 2.3791134 3.3359041 2.6604538 2.7312560 3.3712762 2.9386650
83  3.4589883 2.3176635 2.7900910 2.7561478 2.4882547 3.4221428 2.9679170
84  1.8028654 2.2372766 2.8458862 4.0363096 2.6366486 3.2149336 3.5312969
85  2.6333344 2.1242280 3.1375553 1.2828929 2.2699430 2.2813466 1.7921199
86  1.5984134 1.6162721 2.2705926 3.2785250 1.8011098 2.9003346 2.7805249
87  2.1872085 2.1331610 3.3535848 3.1048150 2.5553013 3.3909524 3.0028579
88  3.1510217 3.0718503 3.6677504 3.9668872 2.8360772 3.9285094 3.2848277
89  2.5709140 2.9653024 3.8697248 4.3832329 3.3254202 3.8159861 3.9015156
90  1.8706564 1.3082827 2.7479826 2.8487311 2.3892578 2.8916881 3.0600461
91  2.2210532 2.2452208 2.7676418 2.6452762 1.7233786 2.3421920 1.7257719
92  2.6804759 2.7769399 3.2474434 3.9810066 2.8106389 3.2115577 3.3307094
93  2.5808057 1.6254837 2.7059715 1.6751770 2.1316687 2.5057048 2.2527783
94  3.0763065 2.7827756 3.7372247 4.1929590 3.1783681 4.2424592 3.9229412
95  2.7143664 2.8524112 3.9664791 4.0393099 3.2330110 3.9205414 3.7067604
96  1.8776540 1.3753725 1.6540620 2.9420073 1.4590577 2.4163543 2.4736442
97  1.5450921 1.3701877 2.3365853 2.7989675 1.8470063 2.3467948 2.5054858
98  2.1359616 2.5030771 2.6110228 4.1285832 2.3249130 3.1567646 3.2461572
99  1.4505707 1.8522206 1.8874775 2.5126860 1.0332048 1.3155002 1.3076471
100 3.2301227 2.6452762 3.1257203 2.5797135 2.3363309 2.8929305 2.2292244
101 2.4373949 2.5282942 3.6975521 3.8173350 3.0042418 3.7341336 3.5539089
102 1.9637955 2.6621555 3.4040404 4.0812311 2.8862897 3.2356997 3.4547747
103 1.1117433 1.4911750 2.1359616 3.2280073 1.9003968 2.3810361 2.7717924
104 2.6281122 2.0145052 2.8480435 3.0809781 2.4097928 3.0308583 2.9575323
105 1.3978452 1.6655187 2.5768581 2.9648746 2.1567854 2.2392820 2.6672596
106 2.9511707 2.4297606 2.9090203 3.9031437 2.5498827 3.7679744 3.4677332
107 2.3044844 2.4802562 3.3427089 3.4796784 2.5143462 3.1296512 2.8721461
108 3.4000041 3.0537355 3.4195255 3.3410062 2.4999452 3.4974650 2.5976617
109 2.4266951 1.6826343 2.2218613 1.5963766 1.6084862 1.9402267 1.6266566
110 3.2892293 2.3386922 2.6968892 2.8561281 2.3471348 3.1784101 2.7944721
111 2.3281775 2.2494341 3.2383288 2.9726855 2.6012979 2.6772374 2.7761332
112 2.8435939 2.7847457 2.8669066 3.6889708 2.2452208 3.0199408 2.7000963
113 2.8487153 2.5030771 2.6110228 3.1526982 2.3249130 2.5322551 2.6428564
114 2.9578487 2.7314283 3.3498649 3.6548729 2.8481919 3.2503534 3.2276801
115 2.5997482 2.1359616 2.4243572 3.7889743 2.2392820 3.4327414 3.2946537
116 2.0832127 1.7938948 3.1486844 2.5556086 2.2797471 3.0459857 2.6070340
117 2.7588340 2.6310005 3.1410624 3.4508775 2.2995033 3.4985314 2.7915868
118 2.2076288 2.5687617 2.1509569 3.4734396 1.5180038 2.3386922 2.0796253
119 2.6711444 2.4491128 2.5756486 3.1741318 1.7857650 2.7850561 2.2090395
120 2.6140070 2.1242280 3.1213516 2.7975638 2.7801091 2.7711893 3.0480855
           71        72        73        74        75        76        77
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72  1.7628702                                                            
73  2.2043353 1.8715678                                                  
74  1.0414878 1.3155941 2.2876962                                        
75  2.6955097 2.1137073 2.1137073 2.6008730                              
76  1.7555938 1.5363754 0.7804271 1.6875278 2.0496666                    
77  1.6084862 1.1706862 2.2075489 1.4617472 2.0154010 1.6746806          
78  2.7512767 1.8874775 2.9692962 1.7938948 2.5465983 2.4429920 2.2210532
79  2.4999452 1.7887773 2.2251084 1.6655187 2.9271513 1.7257719 2.3363309
80  1.8659914 1.4002269 2.9943559 1.2335054 2.8600309 2.3884455 1.2158704
81  2.2494341 2.4728952 2.4728952 1.7170984 3.5191523 1.9115014 2.7220102
82  3.3134291 1.7938948 2.9107056 2.5169667 2.3610709 2.6205314 2.3565131
83  3.1437627 1.8877542 2.9694722 2.5510587 1.8706564 2.8012347 2.4287263
84  2.3679018 2.5363790 3.6658901 1.7356257 4.2404752 3.1400904 2.9352272
85  2.8168762 1.5450921 1.5450921 2.3878886 2.1301856 1.3963204 1.9483337
86  2.1711566 2.0873989 2.8035692 1.6440267 3.5977540 2.5299401 2.8969597
87  3.1305382 2.4450333 2.7802106 2.5819457 4.0137207 2.7050557 3.4447249
88  3.6709303 3.1484624 3.4152889 2.9050614 4.0513324 3.2094693 3.9750167
89  3.3447941 3.0445967 4.0342411 2.4954685 4.6934433 3.4940604 3.5182823
90  2.3443194 1.2961786 2.9471360 1.6152537 3.2994049 2.5200230 1.9846402
91  2.5033798 2.0145052 2.0145052 1.7969372 2.7093543 1.5884767 2.5133541
92  2.8123782 2.5320985 3.6629298 1.8472648 3.6327205 2.9811426 2.6110228
93  2.5567283 0.9133725 2.0825502 2.0155799 1.8776540 1.8277888 1.4976462
94  3.6742779 2.9218963 3.9424624 2.7900910 4.3445898 3.6308476 3.7980811
95  3.5877896 2.9347573 3.7238353 2.7245909 4.5272453 3.3413399 3.6843374
96  1.6241933 1.3638640 2.6672596 0.8076269 2.5610818 2.1803267 1.7867407
97  1.8522206 1.2364476 2.6044100 0.9004469 2.9408631 2.0261824 1.6914846
98  2.3711071 2.7159013 3.5539089 1.6826343 3.9532242 3.0286045 3.0916760
99  1.3701877 1.7555938 1.7555938 1.0169734 2.5227920 1.1202279 1.8676260
100 3.1060183 2.0587076 2.4473782 2.3892209 1.9060594 2.1166109 2.3763358
101 3.3359041 2.6806546 3.5270467 2.5126860 4.3668133 3.1709223 3.4853193
102 2.7666323 2.8072602 3.6242045 2.0564995 4.4456562 3.0594032 3.1722311
103 1.5450921 1.7550755 2.8869428 0.9299284 3.4607205 2.3475071 2.0915112
104 2.6524532 1.5825046 3.0838049 1.6971657 2.7409913 2.5574565 1.9684581
105 1.8278773 1.5573507 2.7711893 1.0883102 3.2795798 2.1304350 1.6826343
106 3.0763065 2.4751262 3.6237800 2.2012098 3.5024207 3.2597106 3.1936781
107 2.9422326 2.3958319 3.0401934 2.0270520 3.7095304 2.5524547 2.9810971
108 3.5388477 2.8329187 2.8329187 2.8151272 2.9810971 2.6645391 3.4838121
109 2.1201411 1.0115880 1.6657411 1.7093117 1.2761367 1.3240603 1.2270309
110 2.9311363 1.8287827 2.9323361 2.2461619 1.8111283 2.6167650 2.1801700
111 2.6489125 1.7857650 2.9057023 1.7687489 3.1213516 2.2652583 1.9060594
112 2.7676418 2.5406567 3.1555827 1.8908327 3.0112840 2.6012979 2.7905652
113 2.3711071 1.9553192 3.0128760 1.6826343 2.2292244 2.3705692 1.5661312
114 3.0080521 2.2962179 3.5040189 2.0219033 3.2064511 2.8655725 2.3909669
115 2.5947363 2.2587457 3.4795781 1.7921199 3.3530036 3.0749443 2.8720832
116 2.9099746 1.9384681 2.3471348 2.3095728 3.4803737 2.2576088 2.9592886
117 3.2440380 2.8342351 2.8342351 2.6820940 3.6956293 2.7823377 3.7310629
118 2.0587076 2.6117140 2.6117140 1.6616028 3.0755463 2.1625827 2.8487153
119 2.6181941 2.2395017 2.6012979 1.8382511 2.5985710 2.1895050 2.6970513
120 2.6281122 1.5450921 3.0647743 1.9619406 2.8355699 2.5054858 1.3939884
           78        79        80        81        82        83        84
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78                                                                       
79  1.4583381                                                            
80  1.8382511 2.1835071                                                  
81  2.3409575 1.2607256 2.5896101                                        
82  1.3155941 1.9846402 2.2813466 3.1675135                              
83  1.7318013 2.6117140 2.4868200 3.5937995 1.2004693                    
84  2.5658142 2.3037585 2.0385034 2.1403377 3.3749814 3.6061621          
85  2.2573943 1.7501024 2.5281523 2.6300695 1.7797197 2.3639501 3.5909049
86  2.4175411 1.9036743 2.4496495 1.9684581 2.9808037 3.0096461 1.4705011
87  2.8720832 1.8811528 3.1675135 2.2813466 2.9911338 3.3716078 2.4243856
88  2.4408171 1.9152859 3.6078398 2.3095728 2.9693934 3.2741134 2.8124280
89  2.5341000 2.0829333 2.6395050 2.1304350 3.2939556 3.8953207 1.3639023
90  1.9762499 1.8970860 1.2538917 2.5498827 2.0887492 2.4798301 1.7501730
91  1.6174516 0.6485839 2.5498827 1.2538917 2.1981067 2.6281122 2.6535828
92  1.3963204 1.8047127 1.7839674 2.0753683 2.5169667 3.0074235 1.8452785
93  1.7687489 1.9115014 1.8450469 2.8955027 1.1342112 1.5053792 3.1975505
94  2.3117123 2.2079103 3.0546986 2.8025366 2.7355365 3.0916760 2.1310827
95  2.4905300 1.8011098 3.0064448 2.2043353 2.9628399 3.6196409 1.9829153
96  1.4500335 1.8278773 1.3044468 2.2090395 2.1671955 2.0155799 1.7867407
97  1.4549339 1.3196460 1.0262418 1.8111283 2.0545330 2.4155768 1.4922957
98  2.2543062 2.1316687 2.3784453 1.8384526 3.3185890 3.4128005 1.0883102
99  2.0437035 1.4075636 2.0753683 1.1313788 2.7587813 2.9013491 2.3411381
100 1.2455510 1.6777891 2.5947363 2.6659387 1.2114344 1.6540620 3.5498629
101 2.4584947 1.7565458 2.8158313 2.1680753 2.8814562 3.4629461 1.7717889
102 2.5959577 1.9442874 2.4175411 1.6174516 3.4317330 3.9326832 1.0270516
103 2.2218613 1.9165305 1.4125673 1.8431487 2.9220654 3.0851576 0.9064062
104 0.7610549 1.6657411 1.3275212 2.5623956 1.2607256 1.7408491 2.2986158
105 1.9545648 1.6241933 1.0115880 1.8028654 2.5391969 2.9779824 1.4822565
106 1.6084862 2.1494648 2.5120991 2.7497916 2.2922540 2.2542514 2.1219552
107 1.7207099 0.9064062 2.4728952 1.4002269 2.4450333 3.0813333 1.9165305
108 1.9238227 1.7662135 3.4647961 2.4746503 2.2776865 2.5066760 3.4967593
109 1.7170984 1.8452785 1.8821664 2.6020963 1.5608542 1.6152537 3.2191119
110 1.2294788 2.3094442 2.1729281 3.2405982 1.0941189 0.6246147 3.3503225
111 1.4308192 1.3240603 1.4419266 2.0445213 1.8692252 2.6801509 2.2985621
112 1.1202279 1.5571480 2.4297606 1.8776540 2.3496539 2.5722447 2.4751262
113 1.2365475 2.1316687 1.4505707 2.6330125 1.9766992 2.1310827 2.8792555
114 0.9004469 1.7338666 1.7233786 2.4230293 1.8389137 2.4680109 2.4201616
115 1.6826343 2.2054997 2.1626398 2.6227817 2.4713907 2.3023508 1.7849789
116 2.4552663 1.4854017 2.7951051 2.1818424 2.4162244 2.8738473 2.5143093
117 2.6555983 1.9716092 3.5561843 2.2616816 3.0459379 3.1483188 2.8589969
118 2.2435265 1.8877542 2.7801091 1.4050577 3.2372539 3.2104569 2.4333226
119 1.2335054 1.3474582 2.5658034 1.8452839 2.1316687 2.2372766 2.6688850
120 1.8287827 2.1941380 1.0178348 2.9269832 1.8079878 2.3853046 2.7457297
           85        86        87        88        89        90        91
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85                                                                       
86  3.0296008                                                            
87  2.7488573 1.4528659                                                  
88  3.2861454 1.9707250 1.8007040                                        
89  3.6002432 2.1097251 2.3412813 2.4228806                              
90  2.3810361 1.8598569 2.1981067 3.0042418 2.1318347                    
91  1.8571888 2.0196550 2.0887492 1.8116369 2.5448547 2.3951221          
92  3.0979726 2.4728952 3.1122620 2.7665988 1.6777891 2.0664096 2.1418940
93  1.1117433 2.7457297 2.8136272 3.3014943 3.4126557 1.7565458 2.1242280
94  3.5044523 1.7969372 1.8470063 1.5266262 1.7463954 2.2218613 2.4911021
95  3.2408163 1.9478633 1.5919117 1.7096594 1.0701978 2.1825769 2.2234715
96  2.5782390 1.5584237 2.5905024 2.6968892 2.5194613 1.5053792 1.9571754
97  2.2079103 1.6084862 2.2222182 2.6401439 1.9036743 0.9004469 1.7429337
98  3.6241671 1.4075636 2.5603671 2.2837397 1.7030386 2.3176635 2.2435265
99  2.1605038 1.9331474 2.5510587 2.7073192 2.7829181 2.3226413 1.2492295
100 1.6336280 3.0306511 3.1284359 2.7378526 3.4607205 2.6871666 1.5584237
101 3.1161151 1.6384038 1.3196460 1.7554141 1.1418698 1.9115014 2.1875287
102 3.4343229 1.8382511 2.2837397 2.5603671 0.7804271 2.0853388 2.3409575
103 2.9003346 1.3383450 2.3207069 2.8929305 1.9060594 1.3383450 2.2292244
104 2.2680317 2.3466610 2.8034184 2.7802106 2.3811088 1.3730377 2.0261824
105 2.4147733 1.9716092 2.5214512 3.0984272 1.9115014 1.1418698 2.0827464
106 3.2994049 1.7298913 2.4560875 1.8659914 2.2874639 2.1301856 2.2680317
107 2.5947781 1.7839674 1.7938948 1.5480251 1.3240603 2.0145052 1.2961786
108 2.5812256 2.6310005 2.5787688 1.4822565 3.2544432 3.1927531 1.3754623
109 1.1899660 2.7064028 2.9995983 3.2818482 3.5568592 2.1339358 1.8499291
110 2.2799378 2.9191973 3.3824245 3.1371348 3.5924530 2.3692405 2.3279725
111 1.9937367 2.5848249 2.7552166 2.9885167 2.0915112 1.5236535 1.8582154
112 2.8693583 2.3133681 2.9554101 2.0691750 2.4248558 2.6214114 1.4397179
113 2.4553806 3.0144533 3.6961025 3.5101510 3.1632316 2.3176635 2.2435265
114 2.6659175 2.7642250 3.2071020 2.8728727 2.1917813 2.0048327 2.0825502
115 3.3104576 1.5117187 2.5629325 2.2292244 2.3087676 1.9571561 2.3212062
116 2.1089039 1.5984134 0.6664127 1.9200627 2.4342768 1.8706564 1.7408491
117 3.0514719 1.6241933 1.4809152 0.9236324 2.8349575 2.9420073 1.7501730
118 3.0769640 1.9202407 2.7847773 2.2813466 2.8355699 2.9555468 1.5283986
119 2.4675759 2.0564995 2.5621882 1.7501730 2.7137950 2.5739091 1.0128552
120 2.1045584 3.0296008 3.3407185 3.7951199 2.9210588 1.4719263 2.6366486
           92        93        94        95        96        97        98
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93  2.7340354                                                            
94  2.3749881 3.1371348                                                  
95  2.2105881 3.1722311 1.1899660                                        
96  1.7857650 1.9309840 2.3892209 2.6561557                              
97  1.4419266 1.7550755 2.2171286 2.0812144 0.9825966                    
98  1.6826343 3.2939556 2.0699773 2.1293704 1.6254837 1.7243896          
99  2.2202831 2.2576088 3.1014456 2.8398898 1.6006835 1.5503690 2.0265761
100 2.4882547 1.5661312 3.0837214 3.1552452 2.2372766 2.2404557 3.1975505
101 2.2461619 2.9921975 1.1213722 0.3981752 2.4383106 1.8811528 2.0145937
102 1.7207099 3.3222954 2.1751959 1.4976462 2.2914097 1.6863972 1.4050577
103 1.8443054 2.5100351 2.4489303 2.2949816 1.2176447 0.9133725 1.3730377
104 1.4158673 1.5450921 2.2543062 2.4266442 1.3155002 1.1202279 2.3117123
105 1.5661312 2.1071376 2.6724660 2.2995033 1.4851893 0.6582843 1.9366493
106 1.9923938 2.7060047 1.2810941 2.1272079 1.5573507 1.8877542 1.7173141
107 1.5645881 2.6010091 1.6144595 1.0470754 2.0587076 1.5236535 1.7408491
108 2.8311341 2.6899915 2.4989369 2.6844255 2.6063654 2.7013407 2.8789308
109 2.6851118 0.8076269 3.3807234 3.3995049 1.7565458 1.8011098 3.1214896
110 2.5126860 1.4809152 2.9995571 3.4293433 1.7501730 2.1256687 3.0903683
111 1.3638640 1.7938948 2.6659387 2.2543062 1.9200627 1.1313788 2.4905300
112 1.3835622 2.6561557 2.3878886 2.4798301 1.7196967 1.8604993 1.7501024
113 1.6826343 1.9350582 3.3749814 3.4117315 1.6254837 1.7243896 2.6656507
114 0.8220133 2.2326288 2.4979786 2.4391040 1.8443054 1.5138224 2.2985621
115 1.8985126 2.6605321 1.6767924 2.3557018 1.1202279 1.6462905 1.3963204
116 2.8903040 2.1926598 1.9635525 1.7257719 2.3191346 1.8989303 2.6456730
117 3.1338001 3.1071066 1.9654637 2.1567854 2.5426962 2.6070340 2.3824163
118 2.3212062 3.0670609 2.9356378 2.8915043 1.8659914 2.1494648 1.6174516
119 1.9652487 2.3386922 2.3409575 2.5040106 1.6540620 1.8860300 1.9846402
120 2.0595794 1.4705011 3.2605553 3.1109336 1.9923688 1.4822565 3.0871352
           99       100       101       102       103       104       105
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99                                                                       
100 2.2171286                                                            
101 2.6967122 3.1300205                                                  
102 2.2776865 3.4469449 1.4437454                                        
103 1.6657411 3.0316479 2.0389056 1.4396997                              
104 2.2431709 1.7338666 2.3265794 2.4713907 1.9384681                    
105 1.6665109 2.6807337 2.1201411 1.5526098 0.8968765 1.5984134          
106 2.6451406 2.5154602 2.0123079 2.4766477 2.1801700 1.6777891 2.4766477
107 1.9684581 2.3265794 1.1202279 1.3963204 1.9188587 1.8676260 1.7938948
108 2.4370977 1.6084862 2.7137950 3.2922158 3.2361014 2.5128467 3.2141265
109 1.7550755 1.3639023 3.2321555 3.3379079 2.4496495 1.7460199 2.1316687
110 2.5764477 1.3082827 3.3117485 3.6127661 2.8382284 1.3638640 2.6624490
111 2.0216672 2.0154010 2.2188630 2.0563445 1.9070010 1.1899660 1.1444998
112 1.8278773 1.8116369 2.5115933 2.3692405 2.2906261 1.7298913 2.2594745
113 2.0265761 1.7659660 3.3412993 3.0132840 2.3319689 1.3383450 1.9366493
114 2.3675648 1.9238227 2.4713907 2.3265794 2.2283133 0.9304698 1.7687489
115 2.3833222 2.6523142 2.1809662 2.3348263 1.7662135 1.6558743 2.1948211
116 2.2749907 2.5842543 1.4783678 2.3789856 2.2229653 2.3745791 2.2417401
117 2.4170864 2.7775457 2.0435491 2.7658804 2.7497527 2.9373214 3.0702635
118 1.1872666 2.5808057 2.8080539 2.4084779 2.1024089 2.6481205 2.4084779
119 1.6082443 1.4865267 2.4721500 2.6088936 2.3428044 1.8384526 2.3799320
120 2.5135119 2.3122551 2.9808037 2.8618858 2.1926598 1.2810941 1.4922957
          106       107       108       109       110       111       112
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104                                                                      
105                                                                      
106                                                                      
107 1.8860300                                                            
108 2.1937743 2.0154010                                                  
109 2.7640523 2.6371024 2.5030771                                        
110 2.0915112 2.7730868 2.3249130 1.4397179                              
111 2.5115933 1.6084862 2.8080539 1.9455697 2.2874639                    
112 1.6971657 1.5784009 1.7204919 2.3602107 2.0853388 2.0123079          
113 2.5499048 2.5658142 2.8789308 1.6241933 1.5635480 1.6278492 1.7501024
114 2.0356695 1.7565458 2.6150000 2.2765560 1.9553192 1.0883102 1.4583381
115 0.6246147 2.0293414 2.5100126 2.6330125 2.1089039 2.4675759 1.7257719
116 2.3639501 1.6655187 2.3060205 2.4266951 2.8865299 2.3174748 2.7206859
117 2.0928951 1.9202407 1.6162721 3.0144533 3.1093216 3.1663617 2.3605990
118 2.3637188 2.0620158 2.1872085 2.5658142 2.8653451 2.7137950 1.4783678
119 1.6746806 1.6239507 1.1418698 2.0025652 1.8756036 2.1861400 0.7804271
120 2.8826563 2.5947781 3.4667409 1.7797197 2.0912040 1.1389480 2.7031124
          113       114       115       116       117       118       119
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113                                                                      
114 1.3155002                                                            
115 2.3457531 2.0595794                                                  
116 3.2484508 2.8398898 2.4747761                                        
117 3.5751365 3.2280073 2.2876962 1.6239507                              
118 2.4837465 2.6366486 2.1418940 2.7038636 2.0860119                    
119 1.9846402 1.9003968 1.7460199 2.2874639 1.8116369 1.3221091          
120 1.5571480 1.6364443 2.7306343 2.8375446 3.8297581 3.3343174 2.8000561
          120       121       122       123       124       125       126
2                                                                        
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          127       128       129       130       131       132       133
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          134       135       136       137       138       139       140
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          141       142       143       144       145       146       147
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          148       149       150       151       152       153       154
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          155       156       157       158       159       160       161
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          162       163       164       165       166       167       168
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          169       170       171       172       173       174       175
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          176       177       178       179       180       181       182
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          183       184       185       186       187       188       189
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          190       191       192       193       194       195       196
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          197       198       199       200       201       202       203
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          204       205       206       207       208       209       210
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          211       212       213       214       215       216       217
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          218       219       220       221       222       223       224
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          225       226       227       228       229       230       231
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          232       233       234       235       236       237       238
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          239       240       241       242       243       244       245
2                                                                        
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          246       247       248       249       250       251       252
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          253       254       255       256       257       258       259
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          260       261       262       263       264       265       266
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          267       268       269       270       271       272       273
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          274       275       276       277       278       279       280
2                                                                        
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          281       282       283       284       285       286       287
2                                                                        
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          288       289       290       291       292       293       294
2                                                                        
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          295       296       297       298       299       300       301
2                                                                        
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          302       303       304       305       306       307       308
2                                                                        
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          309       310       311       312       313       314       315
2                                                                        
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          316       317       318       319       320       321       322
2                                                                        
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120                                                                      
          323       324       325       326       327       328       329
2                                                                        
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120                                                                      
          330       331       332       333       334       335       336
2                                                                        
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120                                                                      
          337       338       339       340       341       342       343
2                                                                        
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          344       345       346       347       348       349       350
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          351       352       353       354       355       356       357
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          358       359       360       361       362       363       364
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          365       366       367       368       369       370       371
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          372       373       374       375       376       377       378
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          379       380       381       382       383       384       385
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          386       387       388       389       390       391       392
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          393       394       395       396       397       398       399
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          400       401       402       403       404       405       406
2                                                                        
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          407       408       409       410       411       412       413
2                                                                        
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120                                                                      
          414       415       416       417       418       419       420
2                                                                        
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          421       422       423       424       425       426       427
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          428       429       430       431       432       433       434
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          435       436       437       438       439       440       441
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          442       443       444       445       446       447       448
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          449       450       451       452       453       454       455
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          456       457       458       459       460       461       462
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          463       464       465       466       467       468       469
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          470       471       472       473       474       475       476
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          477       478       479       480       481       482       483
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          484       485       486       487       488       489       490
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          491       492       493       494       495       496       497
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          498       499       500       501       502       503       504
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          505       506       507       508       509       510       511
2                                                                        
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          512       513       514       515       516       517       518
2                                                                        
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          519       520       521       522       523       524       525
2                                                                        
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          526       527       528       529       530       531       532
2                                                                        
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          533       534       535       536       537       538       539
2                                                                        
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120                                                                      
          540       541       542       543       544       545       546
2                                                                        
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          547       548       549       550       551       552       553
2                                                                        
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120                                                                      
          554       555       556       557       558       559       560
2                                                                        
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          561       562       563       564       565       566       567
2                                                                        
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119                                                                      
120                                                                      
          568       569       570       571       572       573       574
2                                                                        
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111                                                                      
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119                                                                      
120                                                                      
          575       576       577       578       579       580       581
2                                                                        
3                                                                        
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5                                                                        
6                                                                        
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9                                                                        
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100                                                                      
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109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          582       583       584       585       586       587       588
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
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100                                                                      
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104                                                                      
105                                                                      
106                                                                      
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108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          589       590       591       592       593       594       595
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
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100                                                                      
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103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          596       597       598       599       600       601       602
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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100                                                                      
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103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          603       604       605       606       607       608       609
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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15                                                                       
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55                                                                       
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100                                                                      
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103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          610       611       612       613       614       615       616
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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100                                                                      
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103                                                                      
104                                                                      
105                                                                      
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107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          617       618       619       620       621       622       623
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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14                                                                       
15                                                                       
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55                                                                       
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100                                                                      
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103                                                                      
104                                                                      
105                                                                      
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109                                                                      
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111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          624       625       626       627       628       629       630
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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55                                                                       
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80                                                                       
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84                                                                       
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90                                                                       
91                                                                       
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98                                                                       
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100                                                                      
101                                                                      
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103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          631       632       633       634       635       636       637
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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100                                                                      
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103                                                                      
104                                                                      
105                                                                      
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107                                                                      
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109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          638       639       640       641       642       643       644
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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100                                                                      
101                                                                      
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103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
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109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          645       646       647       648       649       650       651
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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100                                                                      
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103                                                                      
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105                                                                      
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107                                                                      
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120                                                                      
          652       653       654       655       656       657       658
2                                                                        
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111                                                                      
112                                                                      
113                                                                      
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115                                                                      
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118                                                                      
119                                                                      
120                                                                      
          659       660       661       662       663       664       665
2                                                                        
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5                                                                        
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100                                                                      
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109                                                                      
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111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          666       667       668       669       670       671       672
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
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100                                                                      
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104                                                                      
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106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          673       674       675       676       677       678       679
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
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10                                                                       
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100                                                                      
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104                                                                      
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106                                                                      
107                                                                      
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109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          680       681       682       683       684       685       686
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
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100                                                                      
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104                                                                      
105                                                                      
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107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          687       688       689       690       691       692       693
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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100                                                                      
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103                                                                      
104                                                                      
105                                                                      
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107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          694       695       696       697       698       699       700
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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100                                                                      
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104                                                                      
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110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          701       702       703       704       705       706       707
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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100                                                                      
101                                                                      
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103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          708       709       710       711       712       713       714
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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100                                                                      
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104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          715       716       717       718       719       720       721
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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55                                                                       
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90                                                                       
91                                                                       
92                                                                       
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94                                                                       
95                                                                       
96                                                                       
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100                                                                      
101                                                                      
102                                                                      
103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          722       723       724       725       726       727       728
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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54                                                                       
55                                                                       
56                                                                       
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74                                                                       
75                                                                       
76                                                                       
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84                                                                       
85                                                                       
86                                                                       
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90                                                                       
91                                                                       
92                                                                       
93                                                                       
94                                                                       
95                                                                       
96                                                                       
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98                                                                       
99                                                                       
100                                                                      
101                                                                      
102                                                                      
103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          729       730       731       732       733       734       735
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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78                                                                       
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80                                                                       
81                                                                       
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84                                                                       
85                                                                       
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90                                                                       
91                                                                       
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100                                                                      
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103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          736       737       738       739       740       741       742
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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16                                                                       
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18                                                                       
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55                                                                       
56                                                                       
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78                                                                       
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80                                                                       
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90                                                                       
91                                                                       
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96                                                                       
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98                                                                       
99                                                                       
100                                                                      
101                                                                      
102                                                                      
103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          743       744       745       746       747       748       749
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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14                                                                       
15                                                                       
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18                                                                       
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54                                                                       
55                                                                       
56                                                                       
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62                                                                       
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76                                                                       
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84                                                                       
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91                                                                       
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100                                                                      
101                                                                      
102                                                                      
103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          750       751       752       753       754       755       756
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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48                                                                       
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50                                                                       
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54                                                                       
55                                                                       
56                                                                       
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62                                                                       
63                                                                       
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65                                                                       
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75                                                                       
76                                                                       
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78                                                                       
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80                                                                       
81                                                                       
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84                                                                       
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91                                                                       
92                                                                       
93                                                                       
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98                                                                       
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100                                                                      
101                                                                      
102                                                                      
103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          757       758       759       760       761       762       763
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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54                                                                       
55                                                                       
56                                                                       
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80                                                                       
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91                                                                       
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98                                                                       
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100                                                                      
101                                                                      
102                                                                      
103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          764       765       766       767       768       769       770
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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54                                                                       
55                                                                       
56                                                                       
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65                                                                       
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93                                                                       
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98                                                                       
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100                                                                      
101                                                                      
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103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          771       772       773       774       775       776       777
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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14                                                                       
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54                                                                       
55                                                                       
56                                                                       
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62                                                                       
63                                                                       
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65                                                                       
66                                                                       
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68                                                                       
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71                                                                       
72                                                                       
73                                                                       
74                                                                       
75                                                                       
76                                                                       
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78                                                                       
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80                                                                       
81                                                                       
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84                                                                       
85                                                                       
86                                                                       
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90                                                                       
91                                                                       
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95                                                                       
96                                                                       
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98                                                                       
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100                                                                      
101                                                                      
102                                                                      
103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          778       779       780       781       782       783       784
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
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13                                                                       
14                                                                       
15                                                                       
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18                                                                       
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25                                                                       
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28                                                                       
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33                                                                       
34                                                                       
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43                                                                       
44                                                                       
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47                                                                       
48                                                                       
49                                                                       
50                                                                       
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52                                                                       
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54                                                                       
55                                                                       
56                                                                       
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61                                                                       
62                                                                       
63                                                                       
64                                                                       
65                                                                       
66                                                                       
67                                                                       
68                                                                       
69                                                                       
70                                                                       
71                                                                       
72                                                                       
73                                                                       
74                                                                       
75                                                                       
76                                                                       
77                                                                       
78                                                                       
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80                                                                       
81                                                                       
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83                                                                       
84                                                                       
85                                                                       
86                                                                       
87                                                                       
88                                                                       
89                                                                       
90                                                                       
91                                                                       
92                                                                       
93                                                                       
94                                                                       
95                                                                       
96                                                                       
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98                                                                       
99                                                                       
100                                                                      
101                                                                      
102                                                                      
103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          785       786       787       788       789       790       791
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
12                                                                       
13                                                                       
14                                                                       
15                                                                       
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18                                                                       
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20                                                                       
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25                                                                       
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27                                                                       
28                                                                       
29                                                                       
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33                                                                       
34                                                                       
35                                                                       
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38                                                                       
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40                                                                       
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42                                                                       
43                                                                       
44                                                                       
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46                                                                       
47                                                                       
48                                                                       
49                                                                       
50                                                                       
51                                                                       
52                                                                       
53                                                                       
54                                                                       
55                                                                       
56                                                                       
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58                                                                       
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60                                                                       
61                                                                       
62                                                                       
63                                                                       
64                                                                       
65                                                                       
66                                                                       
67                                                                       
68                                                                       
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70                                                                       
71                                                                       
72                                                                       
73                                                                       
74                                                                       
75                                                                       
76                                                                       
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78                                                                       
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80                                                                       
81                                                                       
82                                                                       
83                                                                       
84                                                                       
85                                                                       
86                                                                       
87                                                                       
88                                                                       
89                                                                       
90                                                                       
91                                                                       
92                                                                       
93                                                                       
94                                                                       
95                                                                       
96                                                                       
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98                                                                       
99                                                                       
100                                                                      
101                                                                      
102                                                                      
103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          792       793       794       795       796       797       798
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
12                                                                       
13                                                                       
14                                                                       
15                                                                       
16                                                                       
17                                                                       
18                                                                       
19                                                                       
20                                                                       
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22                                                                       
23                                                                       
24                                                                       
25                                                                       
26                                                                       
27                                                                       
28                                                                       
29                                                                       
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33                                                                       
34                                                                       
35                                                                       
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38                                                                       
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40                                                                       
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42                                                                       
43                                                                       
44                                                                       
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46                                                                       
47                                                                       
48                                                                       
49                                                                       
50                                                                       
51                                                                       
52                                                                       
53                                                                       
54                                                                       
55                                                                       
56                                                                       
57                                                                       
58                                                                       
59                                                                       
60                                                                       
61                                                                       
62                                                                       
63                                                                       
64                                                                       
65                                                                       
66                                                                       
67                                                                       
68                                                                       
69                                                                       
70                                                                       
71                                                                       
72                                                                       
73                                                                       
74                                                                       
75                                                                       
76                                                                       
77                                                                       
78                                                                       
79                                                                       
80                                                                       
81                                                                       
82                                                                       
83                                                                       
84                                                                       
85                                                                       
86                                                                       
87                                                                       
88                                                                       
89                                                                       
90                                                                       
91                                                                       
92                                                                       
93                                                                       
94                                                                       
95                                                                       
96                                                                       
97                                                                       
98                                                                       
99                                                                       
100                                                                      
101                                                                      
102                                                                      
103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          799       800       801       802       803       804       805
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
12                                                                       
13                                                                       
14                                                                       
15                                                                       
16                                                                       
17                                                                       
18                                                                       
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25                                                                       
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27                                                                       
28                                                                       
29                                                                       
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33                                                                       
34                                                                       
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40                                                                       
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42                                                                       
43                                                                       
44                                                                       
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46                                                                       
47                                                                       
48                                                                       
49                                                                       
50                                                                       
51                                                                       
52                                                                       
53                                                                       
54                                                                       
55                                                                       
56                                                                       
57                                                                       
58                                                                       
59                                                                       
60                                                                       
61                                                                       
62                                                                       
63                                                                       
64                                                                       
65                                                                       
66                                                                       
67                                                                       
68                                                                       
69                                                                       
70                                                                       
71                                                                       
72                                                                       
73                                                                       
74                                                                       
75                                                                       
76                                                                       
77                                                                       
78                                                                       
79                                                                       
80                                                                       
81                                                                       
82                                                                       
83                                                                       
84                                                                       
85                                                                       
86                                                                       
87                                                                       
88                                                                       
89                                                                       
90                                                                       
91                                                                       
92                                                                       
93                                                                       
94                                                                       
95                                                                       
96                                                                       
97                                                                       
98                                                                       
99                                                                       
100                                                                      
101                                                                      
102                                                                      
103                                                                      
104                                                                      
105                                                                      
106                                                                      
107                                                                      
108                                                                      
109                                                                      
110                                                                      
111                                                                      
112                                                                      
113                                                                      
114                                                                      
115                                                                      
116                                                                      
117                                                                      
118                                                                      
119                                                                      
120                                                                      
          806       807       808       809       810       811       812
2                                                                        
3                                                                        
4                                                                        
5                                                                        
6                                                                        
7                                                                        
8                                                                        
9                                                                        
10                                                                       
11                                                                       
12                                                                       
13                                                                       
14                                                                       
15                                                                       
16                                                                       
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18                                                                       
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20                                                                       
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23                                                                       
24                                                                       
25                                                                       
26                                                                       
27                                                                       
28                                                                       
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 [ reached getOption("max.print") -- omitted 720 rows ]

Does the Data Need Scaling?

As all the ratings are on the same numerical scale, scaling isn’t required. However, scaling can still be used for distance-based clustering because it ensure each drink version contributes equally to the calucations of distance, especially if some drinks have higher variations in ratings than others.

Call:
hclust(d = d_mat, method = "average")

Cluster method   : average 
Distance         : euclidean 
Number of objects: 840 

Dendrogram

Heatmap

Evidence of Clustering Structure in Heatmap

The heatmap does show evidence of a clustering structure, but the clusters are not well defined. There are lighter and darker block patterns along the diagonal, which would suggest that groups of consumers are more similar to each other than to the rest. The boundaries between these blocks are not overly clear, indicating that while preference segments do exist, consumer tastes overlaps to an extent. This would suggest that clustering is appropriate for the data.

  1   2   3 
432 192 216 

Silhouette of 840 units in 3 clusters from silhouette.default(x = clusters_3, dist = d_mat) :
 Cluster sizes and average silhouette widths:
      432       192       216 
0.1789417 0.2177935 0.3168665 
Individual silhouette widths:
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
-0.2758  0.1351  0.2416  0.2233  0.3321  0.5049 

Quality of 3-Cluster Solution

A three-cluster solution results in clusters of 432, 192, and 216. These cluster sizes are well balanced and suitable for segmentation. The silhouette analysis shows average silhouette widths of 0.18, 0.22, and 0.32, with an overall mean of 0.22. This indicated a moderate clustering structure and would suggest that while clusters are reasonably distinct, some overlap in consumer preferences. Overall, the results support the use of this our three-cluster solution.

How Clusters Differ on Age and Gender

Differences in Age and Gender Across Clusters

The clusters are not clearly seperated by gender. All three clusters contain a mix of male and female, and none of the clusters are made up of totally one gender. This suggests that whether someone prefers certain energy drinks is not strongly linked to gender

A similar pattern is seen for age. While there are small differences in how ages are spread across clusters, no cluster is extremely dominant. Each cluster includes a range of ages, meaning that energy drink preferences are shared across different age groups rather than one age being significantly more than others.

Overall, this shows us that customers are grouped mainly based on how they rate the drinks, rather than solely on their age or gender.

Advertising Recommendations

The results suggest that different energy drink versions should be advertised to different clusters based on their preferences. D1, which has the lowest flavour likeness, should be targeted at the cluster that rated lighter flavours more highly, as this group appears to prefer less intense drinks. D3 should be promoted to the largest and most balanced cluster, as it represents a middle option that appeals to a broad range of consumers that fall into the clusters. D5 should be advertised to the cluster that gave higher ratings to stronger flavour, as this would indicate a preference for more intense energy drinks. Targeting each product to the most suitable cluster would improve the impact of marketing campaigns.

Product Recommendations

The one energy drink version that should be continued, D3 would be the most suitable choice. As a medium flavour, this appeals to the widest range of consumers across all clusters, avoiding the lows and the highs of flavour intensity. This makes it the safest option from a brands perspective, as it would maximise overall consumer acceptance and reduce the risk of excluding large segments of the market.