Deep Learning

We will demonstrate how to create deep neural nets with keras, which interfaces with tensorflow. This powerful tool is useful for applications such as automatic postal code recognition, a key feature in logistics systems for accurate deliveries.

library(keras)

## Warning: package 'keras' was built under R version 4.2.3

reticulate::use_condaenv(condaenv = "r-tensorflow")

A Multilayer Network on the MNIST Digit Data

The keras package includes various example datasets, such as the MNIST digit data. This dataset is frequently used for digit classification, which is essential for logistics tasks like reading postal codes.

mnist <- dataset_mnist()
x_train <- mnist$train$x
g_train <- mnist$train$y
x_test <- mnist$test$x
g_test <- mnist$test$y
dim(x_train)

## [1] 60000    28    28

dim(x_test)

## [1] 10000    28    28

The dataset includes 60,000 training images and 10,000 test images, each 28x28 pixels. These are stored in a 3D array, so we need to reshape them into a matrix format. Additionally, we need to “one-hot” encode the labels. Fortunately, keras provides convenient functions to handle this.

x_train <- array_reshape(x_train, c(nrow(x_train), 784))
x_test <- array_reshape(x_test, c(nrow(x_test), 784))
y_train <- to_categorical(g_train, 10)
y_test <- to_categorical(g_test, 10)

Neural networks are sensitive to input scaling. For example, digit values in postal codes (ranging from 0 to 255) must be rescaled to the [0, 1] range for optimal performance.

x_train <- x_train / 255
x_test <- x_test / 255

Now, we are ready to define and fit our neural network model.

modelnn <- keras_model_sequential()
modelnn %>%
  layer_dense(input_shape = c(784), units = 256, activation = "relu") %>%
  layer_dropout(rate = 0.4) %>%
  layer_dense(units = 128, activation = "relu") %>%
  layer_dropout(rate = 0.3) %>%
  layer_dense(units = 10, activation = "softmax")

The first layer converts the 784 input units (from the reshaped images) to a hidden layer with 256 units using the ReLU activation. This structure is typical for tasks like postal code digit recognition in delivery systems.

summary(modelnn)

## Model: "sequential"
## ________________________________________________________________________________
##  Layer (type)                       Output Shape                    Param #     
## ================================================================================
##  dense_2 (Dense)                    (None, 256)                     200960      
##  dropout_1 (Dropout)                (None, 256)                     0           
##  dense_1 (Dense)                    (None, 128)                     32896       
##  dropout (Dropout)                  (None, 128)                     0           
##  dense (Dense)                      (None, 10)                      1290        
## ================================================================================
## Total params: 235146 (918.54 KB)
## Trainable params: 235146 (918.54 KB)
## Non-trainable params: 0 (0.00 Byte)
## ________________________________________________________________________________

The model has 235,146 parameters, including a bias term in each layer. This is essential for fine-tuning the network to perform well on tasks like postal code classification.

Next, we define the fitting process, where we will minimize the cross-entropy loss function, commonly used in classification tasks.

modelnn %>% compile(loss = "categorical_crossentropy",
    optimizer = optimizer_rmsprop(), metrics = c("accuracy")
  )

We are now ready to train the model using the prepared dataset. This step is crucial to teach the model to classify the digits.

system.time(
  history <- modelnn %>%
#     fit(x_train, y_train, epochs = 30, batch_size = 128,
      fit(x_train, y_train, epochs = 15, batch_size = 128,
        validation_split = 0.2)
)

## Epoch 1/15
## 375/375 - 2s - loss: 0.4323 - accuracy: 0.8692 - val_loss: 0.1665 - val_accuracy: 0.9514 - 2s/epoch - 6ms/step
## Epoch 2/15
## 375/375 - 1s - loss: 0.2035 - accuracy: 0.9412 - val_loss: 0.1170 - val_accuracy: 0.9668 - 1s/epoch - 4ms/step
## Epoch 3/15
## 375/375 - 2s - loss: 0.1542 - accuracy: 0.9534 - val_loss: 0.1038 - val_accuracy: 0.9699 - 2s/epoch - 4ms/step
## Epoch 4/15
## 375/375 - 2s - loss: 0.1280 - accuracy: 0.9622 - val_loss: 0.0947 - val_accuracy: 0.9731 - 2s/epoch - 5ms/step
## Epoch 5/15
## 375/375 - 2s - loss: 0.1135 - accuracy: 0.9660 - val_loss: 0.0913 - val_accuracy: 0.9738 - 2s/epoch - 5ms/step
## Epoch 6/15
## 375/375 - 2s - loss: 0.1007 - accuracy: 0.9697 - val_loss: 0.0868 - val_accuracy: 0.9756 - 2s/epoch - 5ms/step
## Epoch 7/15
## 375/375 - 2s - loss: 0.0910 - accuracy: 0.9720 - val_loss: 0.0939 - val_accuracy: 0.9745 - 2s/epoch - 5ms/step
## Epoch 8/15
## 375/375 - 2s - loss: 0.0871 - accuracy: 0.9738 - val_loss: 0.0843 - val_accuracy: 0.9765 - 2s/epoch - 5ms/step
## Epoch 9/15
## 375/375 - 2s - loss: 0.0805 - accuracy: 0.9756 - val_loss: 0.0886 - val_accuracy: 0.9772 - 2s/epoch - 5ms/step
## Epoch 10/15
## 375/375 - 2s - loss: 0.0764 - accuracy: 0.9768 - val_loss: 0.0839 - val_accuracy: 0.9776 - 2s/epoch - 5ms/step
## Epoch 11/15
## 375/375 - 2s - loss: 0.0711 - accuracy: 0.9784 - val_loss: 0.0858 - val_accuracy: 0.9797 - 2s/epoch - 5ms/step
## Epoch 12/15
## 375/375 - 2s - loss: 0.0700 - accuracy: 0.9787 - val_loss: 0.0836 - val_accuracy: 0.9787 - 2s/epoch - 5ms/step
## Epoch 13/15
## 375/375 - 2s - loss: 0.0652 - accuracy: 0.9796 - val_loss: 0.0880 - val_accuracy: 0.9789 - 2s/epoch - 5ms/step
## Epoch 14/15
## 375/375 - 2s - loss: 0.0629 - accuracy: 0.9809 - val_loss: 0.0903 - val_accuracy: 0.9787 - 2s/epoch - 5ms/step
## Epoch 15/15
## 375/375 - 2s - loss: 0.0590 - accuracy: 0.9820 - val_loss: 0.0943 - val_accuracy: 0.9783 - 2s/epoch - 5ms/step

##    user  system elapsed 
##   56.32   11.59   29.19

plot(history, smooth = FALSE)

The validation split of 20% means the model trains on 80% of the data. Training time is essential to ensure the model performs well when applied to logistics systems, especially for digit classification tasks like postal code reading.

We then evaluate the model on the test set to see how well it generalizes.

accuracy <- function(pred, truth)
  mean(drop(as.numeric(pred)) == drop(truth))
modelnn %>% predict(x_test) %>% k_argmax() %>% accuracy(g_test)

## 313/313 - 1s - 597ms/epoch - 2ms/step

## [1] 0.9801

In logistics, it’s important to visualize predictions, such as how well the model can classify postal codes.

modelnn %>% predict(x_test) %>% head() # Probability outputs, with one column showing a 100% probability for digit 7 (postal code digit).

## 313/313 - 1s - 529ms/epoch - 2ms/step

##              [,1]         [,2]         [,3]         [,4]         [,5]
## [1,] 4.720563e-14 1.039206e-12 1.858711e-10 5.073846e-09 1.301160e-15
## [2,] 8.797136e-16 3.766970e-07 9.999996e-01 9.134007e-09 3.048671e-19
## [3,] 8.148786e-10 9.999813e-01 5.949464e-07 1.593997e-07 1.608369e-07
## [4,] 9.999967e-01 5.815658e-11 1.673419e-06 2.277296e-08 4.362244e-08
## [5,] 1.067295e-09 2.633391e-09 8.145318e-10 2.669176e-11 9.999638e-01
## [6,] 4.649016e-12 9.999965e-01 7.068390e-09 4.091315e-09 2.200551e-08
##              [,6]         [,7]         [,8]         [,9]        [,10]
## [1,] 5.265777e-14 3.401260e-19 1.000000e+00 1.658950e-13 1.075909e-11
## [2,] 1.974644e-17 3.408175e-15 1.010883e-11 2.697037e-13 3.644366e-22
## [3,] 4.149212e-08 1.059011e-07 1.284753e-05 4.845840e-06 7.453525e-10
## [4,] 1.043939e-08 1.389175e-06 2.837651e-08 2.064365e-09 1.948252e-07
## [5,] 1.692366e-10 1.058478e-09 3.038299e-07 1.219535e-10 3.588472e-05
## [6,] 1.100398e-10 2.916528e-10 3.336508e-06 1.074491e-07 2.048978e-11

(prognoze <- modelnn %>% predict(x_test) %>% k_argmax() %>% drop() %>% as.numeric())

## 313/313 - 1s - 504ms/epoch - 2ms/step

##     [1] 7 2 1 0 4 1 4 9 5 9 0 6 9 0 1 5 9 7 3 4 9 6 6 5 4 0 7 4 0 1 3 1 3 4 7 2
##    [37] 7 1 2 1 1 7 4 2 3 5 1 2 4 4 6 3 5 5 6 0 4 1 9 5 7 8 9 3 7 4 6 4 3 0 7 0
##    [73] 2 9 1 7 3 2 9 7 7 6 2 7 8 4 7 3 6 1 3 6 9 3 1 4 1 7 6 9 6 0 5 4 9 9 2 1
##   [109] 9 4 8 7 3 9 7 9 4 4 9 2 5 4 7 6 7 9 0 5 8 5 6 6 5 7 8 1 0 1 6 4 6 7 3 1
##   [145] 7 1 8 2 0 2 9 9 5 5 1 5 6 0 3 4 4 6 5 4 6 5 4 5 1 4 4 7 2 3 2 7 1 8 1 8
##   [181] 1 8 5 0 8 9 2 5 0 1 1 1 0 9 0 3 1 6 4 2 3 6 1 1 1 3 9 5 2 9 4 5 9 3 9 0
##   [217] 3 6 5 5 7 2 2 7 1 2 8 4 1 7 3 3 8 8 7 9 2 2 4 1 5 9 8 7 2 3 0 2 4 2 4 1
##   [253] 9 5 7 7 2 8 2 0 8 5 7 7 9 1 8 1 8 0 3 0 1 9 9 4 1 8 2 1 2 9 7 5 9 2 6 4
##   [289] 1 5 8 2 9 2 0 4 0 0 2 8 4 7 1 2 4 0 2 7 4 3 3 0 0 3 1 9 6 5 2 5 9 7 9 3
##   [325] 0 4 2 0 7 1 1 2 1 5 3 3 9 7 8 6 3 6 1 3 8 1 0 5 1 3 1 5 5 6 1 8 5 1 7 9
##   [361] 4 6 2 2 5 0 6 5 6 3 7 2 0 8 8 5 4 1 1 4 0 3 3 7 6 1 6 2 1 9 2 8 6 1 9 5
##   [397] 2 5 4 4 2 8 3 8 2 4 5 0 3 1 7 7 5 7 9 7 1 9 2 1 4 2 9 2 0 4 9 1 4 8 1 8
##   [433] 4 5 9 8 8 3 7 6 0 0 3 0 2 0 6 9 9 3 3 3 2 3 9 1 2 6 8 0 5 6 6 6 3 8 8 2
##   [469] 7 5 8 9 6 1 8 4 1 2 5 9 1 9 7 5 4 0 8 9 9 1 0 5 2 3 7 2 9 4 0 6 3 9 5 2
##   [505] 1 3 1 3 6 5 7 4 2 2 6 3 2 6 5 4 8 9 7 1 3 0 3 8 3 1 9 3 4 4 6 4 2 1 8 2
##   [541] 5 4 8 8 4 0 0 2 3 2 7 7 0 8 7 4 4 7 9 6 9 0 9 8 0 4 6 0 6 3 5 4 8 3 3 9
##   [577] 3 3 3 7 8 0 2 2 1 7 0 6 5 4 3 8 0 9 6 3 8 0 9 9 6 8 6 8 5 7 8 6 0 2 4 0
##   [613] 2 2 3 1 9 7 5 8 0 8 4 6 2 6 7 9 3 2 9 8 2 2 9 2 7 3 5 9 1 8 0 2 0 5 6 1
##   [649] 3 7 6 7 1 2 5 8 0 3 7 2 4 0 9 1 8 6 7 7 4 3 4 9 1 9 3 1 7 3 9 7 6 9 1 3
##   [685] 3 8 3 3 6 7 2 4 5 8 5 1 1 4 4 3 1 0 7 7 0 7 9 4 4 8 5 5 4 0 8 2 1 0 8 4
##   [721] 8 0 4 0 6 1 9 3 2 6 7 2 6 9 3 1 4 6 2 5 9 2 0 6 2 1 7 3 4 1 0 5 4 3 1 1
##   [757] 7 4 9 9 4 8 4 0 2 4 5 1 1 6 4 7 1 9 4 2 4 1 5 5 3 8 3 1 4 5 6 8 9 4 1 5
##   [793] 3 8 0 3 2 5 1 2 8 3 4 4 0 8 8 3 3 1 7 3 5 9 6 3 2 6 1 3 6 0 7 2 1 7 1 4
##   [829] 2 4 2 1 7 9 6 1 1 2 4 8 1 7 7 4 8 0 7 3 1 3 1 0 7 7 0 3 5 5 2 7 6 6 9 2
##   [865] 8 3 5 2 2 5 6 0 8 2 9 2 8 8 8 8 7 4 9 3 0 6 6 3 2 1 3 2 2 9 3 0 0 5 7 8
##   [901] 3 4 4 6 0 2 9 1 4 7 4 7 3 9 8 8 4 7 1 2 1 2 2 3 2 3 2 3 9 1 7 4 0 3 5 5
##   [937] 8 6 3 2 6 7 6 6 3 2 7 9 1 1 7 9 6 4 9 5 1 3 3 4 7 8 9 1 1 0 9 1 4 4 5 4
##   [973] 0 6 2 2 3 1 5 1 2 0 3 8 1 2 6 7 1 6 2 3 9 0 1 2 2 0 8 9 9 0 2 5 1 9 7 8
##  [1009] 1 0 4 1 7 9 5 4 2 6 8 1 3 7 5 4 4 1 8 1 3 8 1 2 5 8 0 6 2 1 1 2 1 5 3 4
##  [1045] 6 9 5 0 9 2 2 4 8 2 1 7 2 4 9 4 4 0 3 9 2 2 3 3 8 3 5 7 3 5 8 1 2 4 4 6
##  [1081] 4 9 5 1 0 6 9 5 9 5 9 7 3 8 0 3 7 1 3 6 7 8 5 9 7 9 6 9 6 3 7 4 6 5 3 5
##  [1117] 4 7 8 7 8 0 7 6 8 8 7 3 3 1 9 5 2 7 3 5 1 1 2 1 4 7 4 7 5 4 5 4 0 8 3 6
##  [1153] 9 6 0 2 7 4 4 4 4 6 6 4 7 9 3 4 5 5 8 7 3 7 2 7 0 2 0 1 1 6 8 9 2 8 7 2
##  [1189] 0 1 5 0 9 1 7 0 6 0 8 6 8 1 8 0 3 3 7 2 3 6 2 1 6 1 1 3 7 9 0 8 0 5 4 0
##  [1225] 2 8 2 2 9 8 4 0 4 5 8 5 1 2 1 3 1 7 9 5 7 2 0 3 8 8 6 2 5 4 1 9 2 1 5 8
##  [1261] 1 0 2 4 4 3 6 8 8 2 4 0 5 0 4 4 7 9 3 4 1 5 9 7 3 5 8 8 0 9 3 3 6 6 0 1
##  [1297] 6 0 3 7 4 4 1 2 9 1 4 6 9 9 3 9 8 4 4 3 1 3 1 3 8 7 9 4 8 8 7 9 9 1 4 5
##  [1333] 6 0 5 2 2 2 1 5 5 2 4 9 6 2 7 7 2 2 1 1 2 8 3 7 2 4 1 7 1 7 6 7 8 2 7 3
##  [1369] 1 7 5 8 2 6 2 2 5 6 6 0 9 2 4 3 3 9 7 6 6 8 0 4 1 3 8 2 9 1 8 0 6 7 2 1
##  [1405] 0 5 5 2 0 2 2 0 2 4 9 8 0 9 9 4 6 5 4 9 1 8 3 4 9 9 1 2 2 8 1 9 6 4 0 9
##  [1441] 4 8 3 8 6 0 2 5 1 9 6 2 9 4 0 9 6 0 6 2 5 4 2 3 8 4 5 5 0 3 8 5 3 5 8 6
##  [1477] 5 7 6 3 3 9 6 1 1 2 9 0 4 3 3 6 9 5 0 3 7 7 7 8 3 9 8 3 0 7 2 7 9 4 5 4
##  [1513] 9 3 2 1 4 0 2 3 7 5 9 8 8 5 0 3 1 4 7 3 9 0 0 0 6 6 2 3 7 8 4 7 7 9 2 4
##  [1549] 1 6 5 2 4 9 9 1 8 4 0 9 8 4 8 7 7 0 7 8 8 6 0 4 8 8 2 4 7 6 6 6 4 7 1 8
##  [1585] 8 2 3 6 3 0 0 3 7 6 9 7 9 9 5 4 3 3 6 1 2 3 7 3 3 2 0 3 3 8 4 3 6 3 5 0
##  [1621] 2 0 9 0 7 4 6 9 3 5 1 9 6 1 4 5 4 5 0 5 9 5 2 1 2 9 1 9 9 4 0 8 4 5 2 9
##  [1657] 2 1 2 1 7 3 6 8 8 4 9 1 9 8 5 7 5 1 1 8 6 5 2 4 4 7 2 3 5 6 8 8 6 2 3 1
##  [1693] 0 5 8 9 2 9 6 7 0 4 8 7 1 7 4 1 0 3 7 2 0 0 9 1 7 0 7 8 4 7 2 0 4 6 0 3
##  [1729] 1 1 3 3 9 6 7 4 1 1 3 0 8 7 3 9 6 9 3 5 0 2 7 2 5 1 2 5 8 0 8 8 1 5 0 3
##  [1765] 0 3 1 4 0 3 7 2 7 1 8 0 7 0 4 3 1 9 8 7 7 1 4 9 9 3 8 1 7 9 0 2 0 3 3 7
##  [1801] 6 9 2 3 3 7 7 0 0 7 5 2 9 8 7 4 4 2 6 6 1 9 6 8 2 9 0 8 3 1 1 6 3 5 1 1
##  [1837] 1 3 1 2 3 0 2 0 1 3 5 5 7 4 3 9 6 9 6 8 3 6 6 8 5 1 4 2 4 4 5 1 1 9 0 2
##  [1873] 4 9 5 7 1 8 3 5 6 9 8 7 1 1 6 7 6 3 2 2 0 8 9 2 5 1 0 8 1 4 5 7 9 6 9 0
##  [1909] 6 1 5 5 8 3 8 2 6 5 0 7 4 6 1 3 4 7 3 2 3 4 2 5 2 7 1 7 2 6 4 1 5 2 8 6
##  [1945] 0 1 8 2 5 7 7 6 9 3 5 8 4 2 4 0 8 8 3 4 9 2 7 5 8 6 5 6 0 8 6 7 3 6 4 9
##  [1981] 4 6 5 3 2 4 1 0 1 4 6 2 9 1 1 0 6 3 9 5 6 5 6 5 3 4 6 4 3 9 1 3 4 1 9 1
##  [2017] 2 1 1 9 3 5 4 0 7 3 6 1 7 5 5 3 3 4 1 3 7 5 8 6 5 1 0 8 2 3 4 6 7 9 8 1
##  [2053] 8 9 9 2 8 6 2 7 0 0 6 7 5 8 6 0 9 3 7 1 3 5 4 3 3 5 5 6 3 0 2 3 4 2 3 0
##  [2089] 9 9 4 7 2 8 4 7 0 6 0 8 5 2 8 5 7 3 0 8 2 7 2 8 2 5 5 7 6 4 0 8 4 8 2 7
##  [2125] 4 5 2 0 3 9 9 6 7 2 5 1 1 1 2 3 6 7 8 7 6 4 8 9 9 8 6 3 8 3 1 0 6 2 2 5
##  [2161] 6 9 8 8 1 4 1 7 8 4 6 1 8 4 3 1 2 8 0 8 5 9 2 4 2 8 2 7 0 7 0 2 5 7 6 7
##  [2197] 9 4 2 6 2 4 4 8 0 4 4 5 8 0 6 8 9 8 5 6 9 0 4 8 7 1 3 4 5 8 0 9 1 3 3 6
##  [2233] 9 8 7 1 0 5 7 1 7 5 2 7 9 1 8 5 2 4 9 4 7 2 2 3 4 9 1 9 2 1 7 9 4 4 1 6
##  [2269] 7 2 7 8 0 1 9 7 1 1 7 5 3 3 5 1 3 7 6 1 3 8 7 3 9 6 0 0 2 8 8 2 3 7 1 3
##  [2305] 0 3 4 4 3 8 9 2 3 9 7 1 1 7 0 4 9 6 5 9 1 7 0 2 0 0 4 6 7 0 7 1 4 6 4 5
##  [2341] 4 9 9 1 7 9 5 3 3 8 2 3 6 2 2 1 1 1 1 1 6 9 8 4 3 7 1 6 4 8 0 4 7 4 2 4
##  [2377] 0 7 0 1 9 8 8 6 0 0 4 1 6 8 2 2 3 8 4 8 2 2 1 7 5 4 4 0 4 3 9 7 3 1 0 1
##  [2413] 2 5 9 2 1 0 1 8 9 1 4 8 3 8 9 3 6 2 8 3 2 2 1 0 4 2 9 2 4 3 7 9 1 5 2 4
##  [2449] 9 0 3 8 5 3 6 0 9 4 6 2 5 0 2 7 4 6 6 8 6 6 8 6 9 1 7 2 5 9 9 0 7 2 7 6
##  [2485] 7 0 6 5 4 4 7 2 0 9 9 2 2 9 4 4 2 3 3 2 1 7 0 7 6 4 1 3 8 7 4 5 9 2 5 1
##  [2521] 8 7 3 7 1 5 3 0 9 1 4 0 6 3 3 6 0 4 9 7 5 1 6 8 9 5 5 7 9 3 8 3 8 1 5 3
##  [2557] 5 0 5 5 3 8 6 7 7 7 3 7 0 5 9 0 2 5 5 3 1 7 7 8 6 5 9 3 8 9 5 3 7 9 1 7
##  [2593] 0 0 3 7 2 3 2 1 8 6 2 9 5 7 5 2 8 6 2 5 1 4 8 4 5 8 3 0 6 2 7 3 3 2 1 0
##  [2629] 7 3 4 0 3 9 3 2 8 9 0 3 8 0 7 6 5 4 7 3 0 0 8 6 2 5 1 1 0 0 4 4 0 1 2 3
##  [2665] 2 7 7 8 5 2 5 7 6 9 1 4 1 6 4 2 4 3 5 4 3 9 5 0 1 5 3 8 9 1 9 7 9 5 5 2
##  [2701] 7 4 6 0 1 1 1 0 4 4 7 6 3 0 0 4 3 0 6 1 9 6 1 3 8 1 2 5 6 2 7 3 6 0 1 9
##  [2737] 7 6 6 8 9 2 9 8 8 3 1 0 0 7 6 6 2 1 6 9 3 1 8 6 9 0 6 0 0 0 6 3 5 9 3 4
##  [2773] 5 5 8 5 3 0 4 0 2 9 6 8 2 3 1 2 1 1 5 6 9 8 0 6 6 5 5 3 8 6 2 1 4 5 4 3
##  [2809] 7 8 3 0 9 3 5 1 1 0 4 4 7 0 1 7 0 1 6 1 4 5 6 6 5 7 8 4 4 7 2 5 3 7 0 7
##  [2845] 7 9 6 4 2 8 5 7 8 3 9 5 8 9 9 8 6 2 8 4 2 3 6 1 1 8 9 3 4 0 7 9 6 7 1 4
##  [2881] 1 3 4 9 3 1 4 7 7 4 7 2 9 3 0 8 8 8 4 0 4 4 1 5 2 8 3 4 9 5 2 8 1 5 3 7
##  [2917] 9 4 2 5 6 2 5 9 3 5 9 2 1 9 5 3 0 6 9 8 4 0 4 7 2 9 0 1 0 3 1 6 5 8 1 5
##  [2953] 3 5 0 3 5 5 9 2 8 7 0 4 9 1 9 7 7 5 3 2 0 9 1 8 6 2 3 9 6 2 1 9 1 3 5 5
##  [2989] 0 3 8 3 3 7 6 6 0 1 4 0 6 9 8 1 2 9 9 5 9 7 3 7 8 0 1 3 0 4 6 1 0 2 5 8
##  [3025] 4 4 1 1 5 4 6 6 0 6 9 2 6 2 7 1 7 9 4 0 0 3 8 2 2 3 1 6 0 5 7 7 9 2 6 7
##  [3061] 7 7 8 6 8 8 4 6 8 4 1 2 8 2 3 9 4 0 3 7 3 2 3 3 7 3 4 0 6 2 0 8 1 5 3 5
##  [3097] 4 1 7 1 5 7 5 7 3 2 2 7 3 7 3 7 8 5 4 5 2 9 6 5 3 6 7 4 1 7 1 5 2 3 6 3
##  [3133] 1 4 2 6 7 4 3 8 0 6 2 1 6 5 3 9 1 9 3 2 1 8 4 4 6 5 8 6 9 7 7 8 6 9 7 3
##  [3169] 9 4 0 5 4 6 4 1 2 3 0 0 2 6 6 5 7 0 8 6 4 7 9 0 7 3 4 2 1 8 8 5 9 2 7 1
##  [3205] 8 8 8 2 7 6 0 1 2 7 1 0 8 3 6 0 5 3 6 2 8 7 0 1 4 2 1 1 4 4 4 4 7 1 6 2
##  [3241] 9 9 0 0 1 8 8 4 3 4 2 0 6 1 6 1 2 2 2 1 2 3 7 8 1 0 0 2 1 6 6 0 1 6 2 5
##  [3277] 1 7 4 8 2 1 4 3 8 3 9 9 4 8 3 4 7 2 7 5 7 0 4 3 3 2 6 7 6 0 0 6 7 7 0 5
##  [3313] 5 8 1 0 7 0 2 8 1 5 0 8 8 0 3 2 7 7 2 6 4 7 5 5 5 2 9 2 8 4 6 8 6 5 0 0
##  [3349] 8 7 6 1 7 1 1 2 7 4 0 0 7 7 6 3 8 6 4 2 0 9 4 0 5 7 8 2 7 4 7 1 1 3 6 6
##  [3385] 2 9 1 9 4 8 3 6 9 5 9 6 2 4 6 7 7 0 6 6 9 9 8 3 5 3 4 9 0 0 5 2 5 0 7 1
##  [3421] 1 1 0 7 6 7 9 6 6 4 1 4 3 1 1 2 2 4 1 0 8 7 6 3 4 0 0 6 3 3 0 7 1 7 1 1
##  [3457] 3 1 0 9 9 7 5 4 1 4 8 9 5 3 5 1 9 8 2 7 3 9 9 0 1 0 2 9 3 9 3 3 6 2 4 9
##  [3493] 8 3 7 4 0 4 7 8 4 9 8 1 9 7 5 9 2 8 2 2 0 2 2 3 8 4 6 8 4 8 2 4 6 7 9 3
##  [3529] 3 9 4 3 1 9 4 7 0 5 9 6 0 4 4 4 4 6 1 2 3 2 6 4 5 9 6 8 5 6 0 8 6 4 1 8
##  [3565] 6 5 2 8 4 5 5 4 7 7 0 7 8 2 2 3 7 0 1 8 0 7 1 9 8 7 5 5 9 1 7 5 4 3 1 2
##  [3601] 2 1 6 6 7 1 1 4 0 7 4 2 4 0 6 4 7 6 9 5 3 4 6 5 0 1 8 8 2 8 3 5 7 8 0 8
##  [3637] 5 7 1 1 0 1 3 7 8 5 0 7 1 1 0 1 1 4 5 2 7 6 2 3 0 2 8 5 9 6 9 7 2 1 3 6
##  [3673] 4 1 8 2 4 0 5 1 0 2 2 6 4 4 3 9 6 1 6 5 7 9 2 0 2 6 0 1 4 3 5 2 8 8 0 8
##  [3709] 8 9 0 9 6 7 6 3 9 3 4 7 7 7 4 9 0 6 4 4 4 2 7 2 8 1 0 0 7 8 3 3 3 1 3 7
##  [3745] 6 1 3 1 6 6 5 2 4 7 5 9 5 8 4 9 9 1 6 5 0 1 3 7 0 3 4 8 2 2 0 2 8 1 5 1
##  [3781] 6 8 8 9 1 2 1 3 5 1 0 9 4 4 8 3 2 5 9 7 6 6 2 0 0 0 5 8 8 1 5 2 3 8 5 1
##  [3817] 8 2 0 4 9 9 6 2 3 3 5 6 4 8 0 9 2 8 3 6 7 5 7 2 9 4 9 1 2 8 6 0 7 0 9 1
##  [3853] 1 5 7 5 9 9 1 9 5 9 2 5 0 4 1 0 8 9 0 8 9 8 9 4 2 5 7 9 8 9 8 0 9 9 6 8
##  [3889] 9 9 5 9 8 6 1 0 3 3 5 2 1 6 5 0 2 8 2 5 6 2 3 0 2 2 6 4 3 5 5 1 7 2 1 6
##  [3925] 9 1 9 9 5 5 1 6 2 2 8 6 7 1 4 6 0 6 0 5 3 2 2 3 6 8 9 8 5 3 8 5 4 5 2 0
##  [3961] 5 6 3 2 8 3 9 9 3 7 9 4 6 7 1 3 7 3 6 6 0 9 0 1 9 4 2 8 8 0 1 6 9 7 5 3
##  [3997] 4 7 4 9 9 4 3 6 3 1 1 7 6 9 1 8 4 1 1 9 9 4 3 6 8 1 6 0 4 1 3 7 7 4 9 5
##  [4033] 1 0 0 1 1 6 2 1 9 8 4 0 3 6 4 9 0 7 1 6 5 7 5 2 5 1 8 5 4 7 0 6 7 2 2 5
##  [4069] 8 1 0 4 5 7 1 3 5 1 9 0 0 6 0 7 3 1 8 3 9 7 0 0 8 9 5 9 8 3 2 7 2 9 7 2
##  [4105] 1 1 3 7 5 3 1 9 8 2 2 2 8 8 5 7 3 8 9 8 8 6 8 2 3 9 7 5 6 2 9 2 8 8 1 6
##  [4141] 2 8 7 9 1 8 0 1 7 2 0 7 5 1 9 0 2 0 9 8 6 2 3 0 3 8 0 2 1 1 1 1 4 2 9 7
##  [4177] 7 5 1 1 2 1 9 9 9 1 0 2 0 2 1 1 4 6 4 1 5 4 9 7 7 1 5 6 2 2 2 8 0 6 9 6
##  [4213] 1 9 7 7 1 4 8 5 3 4 3 4 7 7 5 0 7 4 8 8 1 5 3 9 5 9 7 6 9 0 3 6 3 9 8 2
##  [4249] 8 1 2 8 6 8 5 3 3 9 4 9 2 5 1 5 1 4 4 1 4 4 3 3 9 1 2 2 3 3 0 2 9 0 0 9
##  [4285] 9 6 0 9 3 7 8 4 1 9 9 7 2 7 9 9 5 9 5 1 1 8 3 5 1 9 5 3 5 4 9 5 9 3 1 9
##  [4321] 0 9 7 5 4 9 2 0 1 0 5 1 4 9 3 3 6 1 5 2 5 2 2 0 9 2 6 6 0 1 2 0 3 0 2 5
##  [4357] 5 7 9 5 3 0 8 9 5 0 3 2 5 9 0 8 8 4 5 8 8 4 5 4 8 5 4 9 2 2 1 2 6 8 8 7
##  [4393] 0 3 6 6 4 3 8 8 7 2 2 0 0 9 3 9 9 1 9 8 6 6 4 2 6 9 2 8 5 4 5 7 9 9 9 2
##  [4429] 1 8 3 4 0 7 8 3 9 3 4 6 5 6 2 3 9 2 6 0 0 6 1 2 8 7 9 8 2 0 4 7 7 5 0 5
##  [4465] 6 4 6 7 4 3 0 7 5 0 7 4 2 0 8 9 9 4 2 4 6 7 8 7 6 9 4 1 3 7 3 0 8 7 7 6
##  [4501] 9 3 9 2 2 9 2 1 8 3 2 9 6 8 4 0 1 2 8 4 5 2 7 8 1 1 3 0 3 5 7 0 3 1 9 3
##  [4537] 5 3 1 7 7 3 0 8 4 8 2 6 6 2 9 7 3 9 0 9 9 6 4 2 9 7 2 1 1 6 7 4 7 5 9 2
##  [4573] 8 2 1 4 4 5 7 6 1 3 2 5 9 9 3 6 1 1 4 6 9 7 2 1 5 1 4 6 3 8 1 1 0 3 1 6
##  [4609] 8 4 9 0 7 3 0 2 9 0 6 6 6 3 6 7 7 2 8 6 0 8 3 0 2 9 8 3 2 5 3 8 8 0 0 1
##  [4645] 9 5 1 3 9 6 0 1 4 1 7 1 2 3 7 9 7 4 9 9 3 9 2 8 2 7 1 8 0 9 1 0 1 7 7 9
##  [4681] 6 9 9 9 2 1 6 1 3 5 7 1 9 7 6 4 5 7 6 6 9 9 6 3 6 2 9 8 1 2 2 5 5 2 3 7
##  [4717] 2 1 0 1 0 4 5 2 8 2 8 3 5 1 7 7 1 1 2 9 7 8 4 0 3 0 7 8 8 4 7 7 8 5 8 4
##  [4753] 9 8 1 3 8 0 3 1 7 9 5 5 1 6 5 7 4 9 3 5 4 7 1 2 0 8 1 6 0 7 3 4 7 3 9 6
##  [4789] 0 8 6 4 8 7 7 9 3 8 6 9 7 2 3 4 0 2 1 3 3 5 5 7 2 4 6 7 2 8 3 0 8 7 8 4
##  [4825] 0 8 4 4 5 8 5 6 6 3 0 9 3 7 6 8 9 3 4 9 5 8 9 1 2 8 8 6 8 1 3 7 9 0 1 1
##  [4861] 9 7 0 8 1 7 4 5 7 1 2 1 1 3 9 6 2 1 2 8 8 7 6 6 9 3 7 0 5 2 8 0 5 4 3 8
##  [4897] 4 6 6 2 7 9 5 1 3 2 4 3 6 1 9 4 4 7 6 5 4 1 9 9 2 7 8 0 1 3 6 1 3 4 1 1
##  [4933] 1 5 6 0 7 0 7 2 3 2 5 2 2 9 4 9 8 1 2 1 6 1 2 7 4 0 0 0 8 2 2 9 2 2 7 9
##  [4969] 9 2 7 5 1 3 4 9 4 1 8 5 6 2 8 3 1 2 8 4 9 9 3 7 0 7 7 2 3 2 4 0 3 9 9 8
##  [5005] 4 1 0 6 0 9 6 8 6 1 1 9 8 9 2 3 5 5 9 4 2 1 9 4 3 9 6 0 4 0 6 0 1 2 3 4
##  [5041] 7 8 9 0 1 2 3 4 7 8 9 0 1 2 3 4 5 6 7 8 9 8 3 4 7 8 6 3 4 0 9 7 1 9 3 8
##  [5077] 4 7 3 0 9 1 4 5 4 6 2 0 6 2 1 1 1 1 7 2 4 7 5 2 9 4 5 8 4 2 9 7 0 0 7 5
##  [5113] 1 1 7 6 6 6 8 2 2 7 7 4 0 2 4 2 1 8 9 6 1 0 5 9 6 9 8 0 3 0 8 3 9 6 3 0
##  [5149] 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 5 4 8 7 4 7 7 3 9 8
##  [5185] 8 3 1 5 8 2 7 4 2 1 5 4 5 5 8 6 4 4 4 1 8 7 5 5 1 8 9 1 3 6 3 3 2 2 6 9
##  [5221] 9 6 5 5 3 3 8 1 6 5 6 8 1 9 7 6 8 3 7 4 7 0 9 0 0 3 7 9 3 0 2 0 1 0 1 0
##  [5257] 4 0 1 0 4 7 9 6 2 6 2 2 9 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
##  [5293] 2 3 4 5 6 7 8 9 8 0 5 6 6 0 8 0 2 3 7 9 4 7 1 9 1 7 1 4 0 0 4 1 7 5 7 1
##  [5329] 3 3 3 6 6 9 7 4 3 0 2 5 2 6 0 8 9 4 3 5 4 8 1 5 9 0 6 4 3 6 3 3 8 1 4 7
##  [5365] 5 7 2 2 0 0 1 7 7 9 5 9 8 9 6 8 8 2 3 6 1 2 9 8 9 5 2 6 2 4 8 4 6 5 0 1
##  [5401] 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7 4 2 0 9 0 1 5 8 8 0
##  [5437] 2 7 8 4 4 6 1 0 4 5 3 9 4 2 0 5 0 1 3 2 9 8 6 0 1 1 8 0 4 7 7 6 3 6 0 7
##  [5473] 3 5 4 2 4 1 8 3 5 6 7 0 6 7 1 2 5 8 1 9 3 8 2 8 7 6 7 1 4 6 2 9 3 0 1 2
##  [5509] 3 4 5 6 7 0 1 2 3 4 5 0 1 2 8 9 1 4 0 9 5 0 8 0 7 7 1 1 2 9 3 6 7 2 3 8
##  [5545] 1 2 9 8 8 7 1 7 1 1 0 3 4 2 6 4 7 4 2 7 4 9 1 0 6 8 5 5 5 3 5 9 7 4 8 5
##  [5581] 9 6 9 3 0 3 8 9 1 8 1 6 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2
##  [5617] 3 4 5 6 7 8 9 3 5 3 2 9 3 2 1 4 5 5 2 3 2 1 3 9 7 2 8 2 8 9 1 8 8 7 8 1
##  [5653] 0 0 7 7 8 7 5 0 6 1 5 7 4 6 1 2 5 0 7 9 9 0 3 8 2 4 8 1 8 6 5 9 0 0 0 3
##  [5689] 7 1 6 4 2 6 6 0 4 5 4 1 3 8 6 3 9 9 5 9 3 7 8 5 6 4 7 6 2 2 0 9 4 0 1 2
##  [5725] 3 4 5 6 7 8 9 0 1 2 7 5 6 0 1 2 3 4 5 6 8 7 1 3 2 8 0 7 5 9 9 6 0 9 4 1
##  [5761] 3 2 1 2 3 8 3 2 6 5 6 8 2 7 4 8 1 8 0 5 3 9 4 1 9 2 1 9 6 7 9 0 4 6 1 7
##  [5797] 3 8 7 2 9 6 5 8 3 9 0 5 7 1 6 1 0 9 3 3 4 4 0 6 2 5 4 2 3 4 6 0 0 2 0 1
##  [5833] 4 5 6 7 8 9 0 1 2 3 7 5 6 7 8 0 1 2 3 4 5 6 7 8 9 8 7 1 3 7 5 2 8 0 7 5
##  [5869] 9 9 0 9 1 1 5 8 8 6 3 2 1 8 3 2 6 5 6 0 4 1 0 5 3 1 9 2 1 9 6 0 4 6 1 7
##  [5905] 3 8 7 2 9 6 5 8 3 5 7 1 6 1 0 9 6 2 5 4 2 3 4 4 6 0 0 2 0 1 2 3 9 3 6 7
##  [5941] 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 8 4 5 6 7 8 9 8 6 5 0 6 8 9 4 1 9 3 8 0 4
##  [5977] 8 9 1 4 0 5 5 2 1 5 4 0 7 6 0 1 7 0 6 8 9 5 1 7 9 8 6 0 8 1 7 7 1 3 2 3
##  [6013] 1 4 2 0 0 7 8 4 6 4 9 3 8 4 7 2 3 6 3 6 9 6 3 2 2 4 6 9 0 2 5 5 1 3 3 9
##  [6049] 7 8 7 2 2 5 7 9 8 2 1 3 1 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
##  [6085] 2 3 4 5 6 7 8 9 1 2 6 5 3 0 7 0 4 1 4 3 6 7 2 3 1 2 1 2 9 6 0 1 3 0 2 7
##  [6121] 5 7 6 2 9 1 9 0 6 0 6 0 2 0 6 1 5 8 4 3 0 1 5 4 4 8 5 7 5 7 8 3 4 8 8 5
##  [6157] 2 9 7 1 3 8 1 0 7 5 9 6 9 4 7 7 9 9 3 4 4 3 8 6 2 0 1 2 3 4 5 6 7 8 9 0
##  [6193] 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 8 3 9 5 5 2 6 8 4 9 1 7 1 2 3 5
##  [6229] 9 6 9 1 1 1 2 9 5 6 8 1 2 0 7 7 5 8 2 9 8 9 0 4 6 7 1 3 4 5 6 0 3 6 8 7
##  [6265] 0 4 2 7 4 7 5 4 3 4 2 8 1 5 1 2 0 2 5 6 4 3 0 0 0 3 3 5 7 0 6 4 8 8 6 3
##  [6301] 4 6 9 9 8 2 7 7 1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7
##  [6337] 8 2 1 7 2 5 0 8 0 2 7 8 8 3 6 0 2 7 6 6 1 2 8 8 7 7 4 7 7 3 7 4 5 4 3 3
##  [6373] 8 4 1 1 9 7 4 3 7 3 3 0 2 5 5 6 6 3 5 2 5 9 9 8 4 1 0 6 0 9 6 8 8 5 6 1
##  [6409] 1 9 8 9 2 3 5 5 9 4 2 1 9 2 9 2 0 6 0 4 0 0 1 2 3 4 7 8 9 0 1 2 3 7 8 9
##  [6445] 0 1 2 3 4 7 8 9 7 3 0 3 1 8 7 6 4 0 2 6 8 3 2 8 1 2 0 7 1 0 4 4 5 8 0 6
##  [6481] 2 3 1 5 1 8 5 9 4 0 7 5 8 8 3 8 9 2 6 2 5 3 1 7 3 0 1 9 9 6 0 3 9 2 8 1
##  [6517] 4 3 5 2 9 2 5 8 9 5 0 1 2 4 5 6 0 1 2 3 4 5 6 7 1 2 3 4 5 1 0 4 5 6 6 3
##  [6553] 4 4 2 9 1 0 6 4 9 7 2 3 3 9 2 0 9 3 3 7 1 5 2 3 1 7 8 4 0 2 4 0 2 4 7 8
##  [6589] 0 7 0 6 9 3 2 8 6 7 5 7 5 1 0 8 1 6 7 2 9 7 9 5 8 6 2 6 2 8 1 7 5 0 1 1
##  [6625] 3 7 4 9 1 8 6 8 9 0 1 2 3 4 5 6 7 5 9 0 1 2 3 4 7 8 9 5 1 7 8 9 9 8 9 8
##  [6661] 4 1 7 7 3 3 7 6 6 6 1 9 0 1 7 6 3 2 1 7 1 3 9 1 7 6 8 4 1 4 3 6 9 6 1 4
##  [6697] 4 7 2 4 4 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 9 0 1 2 3 4 7 8 1 3 5 1 7 7
##  [6733] 2 1 4 8 3 4 4 3 9 7 4 1 2 3 5 9 1 6 0 1 0 0 2 7 7 1 1 4 0 4 7 3 6 8 0 3
##  [6769] 7 4 0 6 9 2 6 5 8 6 9 0 4 0 6 6 9 2 0 9 5 1 3 7 6 9 3 0 2 2 0 1 2 3 4 5
##  [6805] 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 2 1 7 2 5 0 8 0 2 7 8 8
##  [6841] 3 0 6 0 2 7 6 4 1 2 8 8 7 7 4 7 7 3 7 4 5 4 3 3 8 4 5 4 1 1 9 7 4 3 7 3
##  [6877] 3 0 2 5 5 6 3 1 5 2 5 9 9 8 4 1 0 6 0 9 6 8 8 5 6 1 1 9 8 9 2 3 5 5 9 4
##  [6913] 2 1 9 4 9 1 3 9 2 0 6 0 4 0 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
##  [6949] 1 2 3 4 5 6 7 8 9 3 8 0 7 1 0 7 5 5 6 9 0 1 0 0 8 3 4 3 1 5 0 0 9 5 3 4
##  [6985] 9 3 7 6 9 2 4 5 7 2 6 4 9 4 9 4 1 2 2 5 8 1 3 2 9 4 3 8 2 2 1 2 8 6 5 1
##  [7021] 6 7 2 1 3 9 3 8 7 5 7 0 7 4 8 8 5 0 6 6 3 7 6 9 9 4 8 4 1 0 6 6 0 1 2 3
##  [7057] 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7 4 0 4 0 1 7 9 5 1
##  [7093] 4 2 8 9 4 3 7 8 2 4 4 3 3 6 9 9 5 8 6 7 0 6 8 2 6 3 9 3 2 8 6 1 7 4 8 8
##  [7129] 9 0 3 3 9 0 5 2 9 4 1 0 3 7 5 8 7 7 8 2 9 7 1 2 6 4 2 5 2 3 6 6 5 0 0 2
##  [7165] 8 1 6 1 0 4 3 1 6 1 9 0 1 4 5 6 7 8 9 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9
##  [7201] 8 4 0 0 7 2 4 3 8 6 6 3 2 6 3 3 3 1 4 7 8 0 3 1 9 0 1 9 1 2 7 0 1 3 8 2
##  [7237] 9 2 7 6 5 5 9 9 8 2 9 1 3 2 3 4 3 1 9 0 9 3 6 8 7 0 1 0 5 8 2 7 7 0 1 2
##  [7273] 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 7 4 8 1 5 6 5 7
##  [7309] 2 8 6 3 3 8 6 5 4 0 9 1 7 2 9 1 5 1 3 2 2 3 0 6 4 3 7 6 9 0 4 8 1 4 0 6
##  [7345] 1 2 6 9 2 2 3 5 5 1 0 7 7 9 6 2 9 4 7 0 2 3 4 0 0 8 8 8 5 1 3 7 4 9 8 8
##  [7381] 9 0 9 8 9 0 2 6 5 6 7 4 7 5 4 1 3 5 3 1 2 3 4 5 6 1 2 3 4 6 0 1 2 4 5 6
##  [7417] 7 8 1 7 2 4 1 4 1 4 9 6 8 4 5 3 7 8 8 3 3 5 6 7 0 6 1 6 8 7 0 1 5 0 8 5
##  [7453] 0 1 5 8 4 2 3 9 7 6 9 1 9 0 6 7 1 2 3 9 2 4 5 5 3 7 5 3 1 8 2 2 3 0 2 9
##  [7489] 4 9 7 0 2 7 4 9 9 2 5 9 8 3 8 6 7 0 0 1 2 3 4 3 6 7 8 9 0 1 2 3 4 5 6 7
##  [7525] 8 9 0 1 2 3 4 5 6 7 8 9 0 0 7 2 6 5 5 3 7 8 6 6 6 6 4 3 8 8 3 0 1 9 0 5
##  [7561] 4 1 9 1 2 7 0 1 3 8 2 9 2 7 4 2 6 5 5 9 9 1 1 5 7 6 8 2 9 4 3 1 9 0 9 3
##  [7597] 6 8 7 0 1 0 5 8 2 7 7 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 8 9 0 1 2 3 4 5 6
##  [7633] 7 8 9 2 1 2 1 3 9 9 8 5 3 7 0 7 7 5 7 9 9 4 7 0 3 4 1 5 8 1 4 8 4 1 8 6
##  [7669] 6 4 6 0 5 5 3 3 5 7 2 5 9 6 9 2 6 2 1 2 0 8 3 8 3 0 8 7 4 9 5 0 9 7 0 0
##  [7705] 4 6 0 9 1 6 2 7 6 8 3 5 2 1 8 3 8 6 1 0 2 1 4 0 1 2 3 4 5 6 7 8 9 0 1 2
##  [7741] 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7 6 4 7 6 2 3 4 8 7 8 6 9 8 3 2 2 8 4
##  [7777] 8 5 6 5 0 2 0 1 1 2 9 6 8 2 1 0 6 5 2 9 7 5 3 9 2 7 1 8 3 8 1 9 5 5 0 1
##  [7813] 8 9 8 2 6 0 4 5 0 2 1 8 6 7 5 9 9 3 0 3 1 4 4 0 4 9 0 1 2 3 5 6 7 8 0 1
##  [7849] 2 3 5 6 7 8 9 0 1 2 3 5 6 7 8 9 9 7 0 9 0 1 5 8 8 0 9 3 2 7 8 4 6 1 0 4
##  [7885] 9 4 2 0 5 0 1 6 9 3 2 9 1 6 0 8 1 8 7 7 6 3 6 0 7 2 4 1 7 0 6 7 1 2 5 8
##  [7921] 1 2 2 8 7 6 8 7 1 6 2 9 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2
##  [7957] 3 4 5 6 7 8 9 8 9 5 7 0 3 1 6 8 4 1 5 6 4 2 7 8 1 3 4 3 4 7 2 0 5 0 8 9
##  [7993] 2 3 2 3 5 5 7 8 4 9 9 7 1 1 9 0 7 8 3 4 8 6 3 8 0 9 6 2 1 0 1 0 6 2 3 8
##  [8029] 9 0 7 2 3 4 5 5 2 8 5 4 6 6 6 7 9 1 8 2 1 5 3 4 7 9 4 0 0 0 1 2 3 4 5 6
##  [8065] 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 9 0 1 3 1 5 1 2 4 9 8 4 6 8 0 1
##  [8101] 1 9 2 6 6 8 7 4 2 9 7 0 2 1 0 3 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8
##  [8137] 9 0 1 2 3 4 5 6 7 8 9 8 6 5 9 7 0 2 3 4 3 8 5 1 5 2 3 0 1 2 1 3 2 6 5 3
##  [8173] 0 7 2 7 4 6 4 0 5 9 9 8 9 5 3 1 7 4 7 6 5 4 0 0 6 6 2 0 6 3 7 7 4 4 3 9
##  [8209] 2 8 9 6 0 9 5 3 8 8 7 1 4 0 4 8 5 2 3 9 0 1 9 1 5 1 7 4 8 6 2 1 6 8 8 0
##  [8245] 1 2 9 4 7 8 9 0 1 2 3 4 6 7 8 9 0 1 2 3 4 7 8 9 1 4 5 3 3 0 9 5 4 9 0 8
##  [8281] 4 6 7 0 7 7 1 6 9 1 3 6 2 3 8 2 3 8 9 5 8 8 7 1 7 1 1 0 3 4 2 4 4 7 4 2
##  [8317] 7 4 2 9 2 7 9 2 1 6 6 5 3 4 8 5 9 6 9 0 6 3 0 8 1 6 0 0 1 2 3 4 5 6 7 0
##  [8353] 1 2 3 4 7 8 9 0 1 2 3 4 7 2 5 1 6 4 3 9 9 0 9 7 1 6 4 3 6 2 0 9 8 6 5 7
##  [8389] 0 0 1 7 4 3 2 4 1 3 7 6 4 7 7 7 9 8 4 3 6 2 8 3 5 8 0 5 4 7 1 3 1 7 9 6
##  [8425] 2 0 9 1 7 3 3 9 1 6 4 3 9 8 2 1 8 6 4 1 5 5 6 5 0 1 2 3 4 5 6 7 8 9 0 1
##  [8461] 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 6 9 7 0 2 3 4 3 8 5 1 3 0 1 2 1 3 2
##  [8497] 0 7 2 6 4 0 5 9 9 8 9 5 3 1 7 4 7 0 0 6 6 6 3 7 9 2 8 9 8 7 1 9 0 4 8 5
##  [8533] 2 3 9 0 1 9 1 5 1 7 6 1 2 1 6 8 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 0
##  [8569] 1 2 3 5 6 7 8 1 0 4 5 6 6 3 4 4 2 8 1 0 6 4 9 7 2 9 2 0 9 3 3 9 1 5 2 3
##  [8605] 1 6 7 3 7 8 4 0 2 4 0 2 4 7 8 0 7 0 6 9 3 2 4 8 6 0 5 7 5 1 0 8 1 6 7 2
##  [8641] 9 7 9 5 6 5 2 6 2 8 1 7 5 5 7 3 5 0 1 1 3 8 4 9 4 5 1 8 6 8 9 0 1 2 3 4
##  [8677] 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 3 5 3 2 9 3 2 1 4 5 5
##  [8713] 2 3 2 1 3 9 7 2 1 2 8 9 1 8 8 7 8 1 0 0 6 7 7 8 7 5 0 6 1 5 7 4 6 1 2 5
##  [8749] 0 7 9 9 0 3 4 4 8 4 1 8 6 5 9 0 0 0 3 7 1 6 4 6 0 4 5 4 1 3 8 6 3 9 9 5
##  [8785] 9 3 7 8 5 6 4 7 6 2 2 0 9 4 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
##  [8821] 2 3 4 5 6 7 8 9 6 4 2 6 4 7 5 5 4 7 2 9 3 9 3 8 2 0 9 5 6 0 1 0 6 5 3 5
##  [8857] 3 8 0 0 3 4 1 5 3 0 8 3 0 6 2 7 8 1 7 1 3 8 5 4 2 0 9 7 6 7 4 1 6 2 6 7
##  [8893] 1 9 8 0 6 9 4 9 9 6 2 3 7 1 9 2 2 5 3 7 8 0 1 2 3 4 7 8 9 0 1 2 3 4 7 8
##  [8929] 9 0 1 7 8 9 8 9 2 6 1 3 5 4 8 2 6 4 3 4 5 9 2 0 3 9 4 9 7 3 8 7 4 4 9 8
##  [8965] 5 8 2 6 6 2 3 1 3 2 7 3 1 9 0 1 1 3 5 0 7 8 1 5 1 4 6 0 0 4 9 1 6 6 9 0
##  [9001] 7 6 1 1 0 1 2 3 4 2 2 3 4 5 6 2 0 1 2 2 8 6 3 9 2 1 9 3 9 6 1 7 2 4 4 5
##  [9037] 7 0 0 1 6 6 8 2 7 7 2 4 2 1 6 1 0 6 9 8 3 9 6 3 0 1 2 3 4 5 6 7 8 9 0 1
##  [9073] 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 6 8 9 9 0 1 2 4 4 3 7 4 4 4 0 3 8
##  [9109] 7 5 8 2 1 7 5 3 8 5 2 5 1 1 6 2 1 3 8 6 4 2 6 2 5 5 0 2 8 0 6 8 1 7 9 1
##  [9145] 9 2 6 7 6 6 8 7 4 9 2 1 3 3 0 5 5 8 0 3 7 9 7 0 2 7 9 1 7 8 0 3 5 3 6 0
##  [9181] 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 7 8 9 6 4 2 6 4 7 8 9 2
##  [9217] 9 3 9 3 0 0 1 0 4 2 6 3 5 3 0 3 4 1 5 3 0 8 3 0 6 1 7 8 0 9 2 6 7 1 9 6
##  [9253] 9 4 9 9 6 7 1 2 5 3 7 8 0 1 2 4 5 6 7 8 9 0 1 3 4 5 6 7 5 0 1 3 4 7 8 9
##  [9289] 7 5 5 1 9 9 7 1 0 0 5 9 7 1 7 2 2 3 6 8 3 2 0 0 6 1 7 5 8 6 2 9 4 8 8 7
##  [9325] 1 0 8 7 7 5 8 5 3 4 6 1 1 5 5 0 7 2 3 6 4 1 2 4 1 5 4 2 0 4 8 6 1 9 0 2
##  [9361] 5 6 9 3 6 3 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 5 6 7 8 1
##  [9397] 0 9 5 7 5 1 8 6 9 0 4 1 9 3 8 4 4 7 0 1 9 2 8 7 8 2 5 9 6 0 6 5 5 3 3 3
##  [9433] 9 8 1 1 0 6 1 0 0 6 2 1 1 3 2 7 7 8 8 7 8 4 6 0 2 0 7 0 3 6 8 7 1 5 9 9
##  [9469] 3 7 2 4 9 4 3 6 2 2 5 3 2 5 5 9 4 1 7 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
##  [9505] 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 0 1 2 7 5 3 4 4 0 0 6 9 6 6 5 7 2 3 4 4 9
##  [9541] 1 4 0 7 9 5 7 2 3 1 4 4 0 9 9 6 1 8 3 3 7 3 9 8 8 4 7 7 6 2 1 9 8 7 8 8
##  [9577] 7 2 2 3 9 3 3 5 5 0 7 4 5 6 5 1 4 1 1 2 8 2 6 1 5 0 1 2 3 4 5 6 7 8 9 0
##  [9613] 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 8 0 6 0 2 2 3 7 9 4 7 1 9 1 7 1 4 0
##  [9649] 0 1 7 5 7 1 3 3 3 1 6 9 7 1 3 0 7 6 0 8 9 7 3 5 4 8 1 5 9 0 6 3 3 8 1 4
##  [9685] 7 5 2 0 0 1 7 8 9 6 8 8 2 3 6 1 2 9 5 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
##  [9721] 6 7 8 9 0 1 2 3 4 6 6 7 8 9 7 4 6 1 4 0 9 9 3 7 8 0 7 5 8 5 3 2 2 0 5 8
##  [9757] 6 0 3 8 1 0 3 0 4 7 4 9 0 9 0 7 1 7 1 6 6 0 6 2 8 7 6 4 9 9 5 3 7 4 3 0
##  [9793] 9 6 6 1 1 3 2 1 0 0 1 2 3 4 7 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 7 8 9 0 8
##  [9829] 3 9 5 5 2 6 8 4 1 7 1 7 3 5 6 9 1 1 1 2 1 2 0 7 7 5 8 2 9 8 6 7 3 4 6 8
##  [9865] 7 0 4 2 7 7 5 4 3 4 2 8 1 5 1 0 2 3 3 5 7 0 6 8 6 3 9 9 8 2 7 7 1 0 1 7
##  [9901] 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 7 8 9 7 8 6 4 1 9 3 8 4 4 7 0 1 9 2 8 7
##  [9937] 8 2 6 0 6 5 3 3 8 9 1 4 0 6 1 0 0 6 2 1 1 7 7 8 4 6 0 7 0 3 6 8 7 1 5 2
##  [9973] 4 9 4 3 6 4 1 7 2 6 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6

Comparing the predicted labels with the true labels helps assess how well the model is classifying postal codes.

prognoze

##     [1] 7 2 1 0 4 1 4 9 5 9 0 6 9 0 1 5 9 7 3 4 9 6 6 5 4 0 7 4 0 1 3 1 3 4 7 2
##    [37] 7 1 2 1 1 7 4 2 3 5 1 2 4 4 6 3 5 5 6 0 4 1 9 5 7 8 9 3 7 4 6 4 3 0 7 0
##    [73] 2 9 1 7 3 2 9 7 7 6 2 7 8 4 7 3 6 1 3 6 9 3 1 4 1 7 6 9 6 0 5 4 9 9 2 1
##   [109] 9 4 8 7 3 9 7 9 4 4 9 2 5 4 7 6 7 9 0 5 8 5 6 6 5 7 8 1 0 1 6 4 6 7 3 1
##   [145] 7 1 8 2 0 2 9 9 5 5 1 5 6 0 3 4 4 6 5 4 6 5 4 5 1 4 4 7 2 3 2 7 1 8 1 8
##   [181] 1 8 5 0 8 9 2 5 0 1 1 1 0 9 0 3 1 6 4 2 3 6 1 1 1 3 9 5 2 9 4 5 9 3 9 0
##   [217] 3 6 5 5 7 2 2 7 1 2 8 4 1 7 3 3 8 8 7 9 2 2 4 1 5 9 8 7 2 3 0 2 4 2 4 1
##   [253] 9 5 7 7 2 8 2 0 8 5 7 7 9 1 8 1 8 0 3 0 1 9 9 4 1 8 2 1 2 9 7 5 9 2 6 4
##   [289] 1 5 8 2 9 2 0 4 0 0 2 8 4 7 1 2 4 0 2 7 4 3 3 0 0 3 1 9 6 5 2 5 9 7 9 3
##   [325] 0 4 2 0 7 1 1 2 1 5 3 3 9 7 8 6 3 6 1 3 8 1 0 5 1 3 1 5 5 6 1 8 5 1 7 9
##   [361] 4 6 2 2 5 0 6 5 6 3 7 2 0 8 8 5 4 1 1 4 0 3 3 7 6 1 6 2 1 9 2 8 6 1 9 5
##   [397] 2 5 4 4 2 8 3 8 2 4 5 0 3 1 7 7 5 7 9 7 1 9 2 1 4 2 9 2 0 4 9 1 4 8 1 8
##   [433] 4 5 9 8 8 3 7 6 0 0 3 0 2 0 6 9 9 3 3 3 2 3 9 1 2 6 8 0 5 6 6 6 3 8 8 2
##   [469] 7 5 8 9 6 1 8 4 1 2 5 9 1 9 7 5 4 0 8 9 9 1 0 5 2 3 7 2 9 4 0 6 3 9 5 2
##   [505] 1 3 1 3 6 5 7 4 2 2 6 3 2 6 5 4 8 9 7 1 3 0 3 8 3 1 9 3 4 4 6 4 2 1 8 2
##   [541] 5 4 8 8 4 0 0 2 3 2 7 7 0 8 7 4 4 7 9 6 9 0 9 8 0 4 6 0 6 3 5 4 8 3 3 9
##   [577] 3 3 3 7 8 0 2 2 1 7 0 6 5 4 3 8 0 9 6 3 8 0 9 9 6 8 6 8 5 7 8 6 0 2 4 0
##   [613] 2 2 3 1 9 7 5 8 0 8 4 6 2 6 7 9 3 2 9 8 2 2 9 2 7 3 5 9 1 8 0 2 0 5 6 1
##   [649] 3 7 6 7 1 2 5 8 0 3 7 2 4 0 9 1 8 6 7 7 4 3 4 9 1 9 3 1 7 3 9 7 6 9 1 3
##   [685] 3 8 3 3 6 7 2 4 5 8 5 1 1 4 4 3 1 0 7 7 0 7 9 4 4 8 5 5 4 0 8 2 1 0 8 4
##   [721] 8 0 4 0 6 1 9 3 2 6 7 2 6 9 3 1 4 6 2 5 9 2 0 6 2 1 7 3 4 1 0 5 4 3 1 1
##   [757] 7 4 9 9 4 8 4 0 2 4 5 1 1 6 4 7 1 9 4 2 4 1 5 5 3 8 3 1 4 5 6 8 9 4 1 5
##   [793] 3 8 0 3 2 5 1 2 8 3 4 4 0 8 8 3 3 1 7 3 5 9 6 3 2 6 1 3 6 0 7 2 1 7 1 4
##   [829] 2 4 2 1 7 9 6 1 1 2 4 8 1 7 7 4 8 0 7 3 1 3 1 0 7 7 0 3 5 5 2 7 6 6 9 2
##   [865] 8 3 5 2 2 5 6 0 8 2 9 2 8 8 8 8 7 4 9 3 0 6 6 3 2 1 3 2 2 9 3 0 0 5 7 8
##   [901] 3 4 4 6 0 2 9 1 4 7 4 7 3 9 8 8 4 7 1 2 1 2 2 3 2 3 2 3 9 1 7 4 0 3 5 5
##   [937] 8 6 3 2 6 7 6 6 3 2 7 9 1 1 7 9 6 4 9 5 1 3 3 4 7 8 9 1 1 0 9 1 4 4 5 4
##   [973] 0 6 2 2 3 1 5 1 2 0 3 8 1 2 6 7 1 6 2 3 9 0 1 2 2 0 8 9 9 0 2 5 1 9 7 8
##  [1009] 1 0 4 1 7 9 5 4 2 6 8 1 3 7 5 4 4 1 8 1 3 8 1 2 5 8 0 6 2 1 1 2 1 5 3 4
##  [1045] 6 9 5 0 9 2 2 4 8 2 1 7 2 4 9 4 4 0 3 9 2 2 3 3 8 3 5 7 3 5 8 1 2 4 4 6
##  [1081] 4 9 5 1 0 6 9 5 9 5 9 7 3 8 0 3 7 1 3 6 7 8 5 9 7 9 6 9 6 3 7 4 6 5 3 5
##  [1117] 4 7 8 7 8 0 7 6 8 8 7 3 3 1 9 5 2 7 3 5 1 1 2 1 4 7 4 7 5 4 5 4 0 8 3 6
##  [1153] 9 6 0 2 7 4 4 4 4 6 6 4 7 9 3 4 5 5 8 7 3 7 2 7 0 2 0 1 1 6 8 9 2 8 7 2
##  [1189] 0 1 5 0 9 1 7 0 6 0 8 6 8 1 8 0 3 3 7 2 3 6 2 1 6 1 1 3 7 9 0 8 0 5 4 0
##  [1225] 2 8 2 2 9 8 4 0 4 5 8 5 1 2 1 3 1 7 9 5 7 2 0 3 8 8 6 2 5 4 1 9 2 1 5 8
##  [1261] 1 0 2 4 4 3 6 8 8 2 4 0 5 0 4 4 7 9 3 4 1 5 9 7 3 5 8 8 0 9 3 3 6 6 0 1
##  [1297] 6 0 3 7 4 4 1 2 9 1 4 6 9 9 3 9 8 4 4 3 1 3 1 3 8 7 9 4 8 8 7 9 9 1 4 5
##  [1333] 6 0 5 2 2 2 1 5 5 2 4 9 6 2 7 7 2 2 1 1 2 8 3 7 2 4 1 7 1 7 6 7 8 2 7 3
##  [1369] 1 7 5 8 2 6 2 2 5 6 6 0 9 2 4 3 3 9 7 6 6 8 0 4 1 3 8 2 9 1 8 0 6 7 2 1
##  [1405] 0 5 5 2 0 2 2 0 2 4 9 8 0 9 9 4 6 5 4 9 1 8 3 4 9 9 1 2 2 8 1 9 6 4 0 9
##  [1441] 4 8 3 8 6 0 2 5 1 9 6 2 9 4 0 9 6 0 6 2 5 4 2 3 8 4 5 5 0 3 8 5 3 5 8 6
##  [1477] 5 7 6 3 3 9 6 1 1 2 9 0 4 3 3 6 9 5 0 3 7 7 7 8 3 9 8 3 0 7 2 7 9 4 5 4
##  [1513] 9 3 2 1 4 0 2 3 7 5 9 8 8 5 0 3 1 4 7 3 9 0 0 0 6 6 2 3 7 8 4 7 7 9 2 4
##  [1549] 1 6 5 2 4 9 9 1 8 4 0 9 8 4 8 7 7 0 7 8 8 6 0 4 8 8 2 4 7 6 6 6 4 7 1 8
##  [1585] 8 2 3 6 3 0 0 3 7 6 9 7 9 9 5 4 3 3 6 1 2 3 7 3 3 2 0 3 3 8 4 3 6 3 5 0
##  [1621] 2 0 9 0 7 4 6 9 3 5 1 9 6 1 4 5 4 5 0 5 9 5 2 1 2 9 1 9 9 4 0 8 4 5 2 9
##  [1657] 2 1 2 1 7 3 6 8 8 4 9 1 9 8 5 7 5 1 1 8 6 5 2 4 4 7 2 3 5 6 8 8 6 2 3 1
##  [1693] 0 5 8 9 2 9 6 7 0 4 8 7 1 7 4 1 0 3 7 2 0 0 9 1 7 0 7 8 4 7 2 0 4 6 0 3
##  [1729] 1 1 3 3 9 6 7 4 1 1 3 0 8 7 3 9 6 9 3 5 0 2 7 2 5 1 2 5 8 0 8 8 1 5 0 3
##  [1765] 0 3 1 4 0 3 7 2 7 1 8 0 7 0 4 3 1 9 8 7 7 1 4 9 9 3 8 1 7 9 0 2 0 3 3 7
##  [1801] 6 9 2 3 3 7 7 0 0 7 5 2 9 8 7 4 4 2 6 6 1 9 6 8 2 9 0 8 3 1 1 6 3 5 1 1
##  [1837] 1 3 1 2 3 0 2 0 1 3 5 5 7 4 3 9 6 9 6 8 3 6 6 8 5 1 4 2 4 4 5 1 1 9 0 2
##  [1873] 4 9 5 7 1 8 3 5 6 9 8 7 1 1 6 7 6 3 2 2 0 8 9 2 5 1 0 8 1 4 5 7 9 6 9 0
##  [1909] 6 1 5 5 8 3 8 2 6 5 0 7 4 6 1 3 4 7 3 2 3 4 2 5 2 7 1 7 2 6 4 1 5 2 8 6
##  [1945] 0 1 8 2 5 7 7 6 9 3 5 8 4 2 4 0 8 8 3 4 9 2 7 5 8 6 5 6 0 8 6 7 3 6 4 9
##  [1981] 4 6 5 3 2 4 1 0 1 4 6 2 9 1 1 0 6 3 9 5 6 5 6 5 3 4 6 4 3 9 1 3 4 1 9 1
##  [2017] 2 1 1 9 3 5 4 0 7 3 6 1 7 5 5 3 3 4 1 3 7 5 8 6 5 1 0 8 2 3 4 6 7 9 8 1
##  [2053] 8 9 9 2 8 6 2 7 0 0 6 7 5 8 6 0 9 3 7 1 3 5 4 3 3 5 5 6 3 0 2 3 4 2 3 0
##  [2089] 9 9 4 7 2 8 4 7 0 6 0 8 5 2 8 5 7 3 0 8 2 7 2 8 2 5 5 7 6 4 0 8 4 8 2 7
##  [2125] 4 5 2 0 3 9 9 6 7 2 5 1 1 1 2 3 6 7 8 7 6 4 8 9 9 8 6 3 8 3 1 0 6 2 2 5
##  [2161] 6 9 8 8 1 4 1 7 8 4 6 1 8 4 3 1 2 8 0 8 5 9 2 4 2 8 2 7 0 7 0 2 5 7 6 7
##  [2197] 9 4 2 6 2 4 4 8 0 4 4 5 8 0 6 8 9 8 5 6 9 0 4 8 7 1 3 4 5 8 0 9 1 3 3 6
##  [2233] 9 8 7 1 0 5 7 1 7 5 2 7 9 1 8 5 2 4 9 4 7 2 2 3 4 9 1 9 2 1 7 9 4 4 1 6
##  [2269] 7 2 7 8 0 1 9 7 1 1 7 5 3 3 5 1 3 7 6 1 3 8 7 3 9 6 0 0 2 8 8 2 3 7 1 3
##  [2305] 0 3 4 4 3 8 9 2 3 9 7 1 1 7 0 4 9 6 5 9 1 7 0 2 0 0 4 6 7 0 7 1 4 6 4 5
##  [2341] 4 9 9 1 7 9 5 3 3 8 2 3 6 2 2 1 1 1 1 1 6 9 8 4 3 7 1 6 4 8 0 4 7 4 2 4
##  [2377] 0 7 0 1 9 8 8 6 0 0 4 1 6 8 2 2 3 8 4 8 2 2 1 7 5 4 4 0 4 3 9 7 3 1 0 1
##  [2413] 2 5 9 2 1 0 1 8 9 1 4 8 3 8 9 3 6 2 8 3 2 2 1 0 4 2 9 2 4 3 7 9 1 5 2 4
##  [2449] 9 0 3 8 5 3 6 0 9 4 6 2 5 0 2 7 4 6 6 8 6 6 8 6 9 1 7 2 5 9 9 0 7 2 7 6
##  [2485] 7 0 6 5 4 4 7 2 0 9 9 2 2 9 4 4 2 3 3 2 1 7 0 7 6 4 1 3 8 7 4 5 9 2 5 1
##  [2521] 8 7 3 7 1 5 3 0 9 1 4 0 6 3 3 6 0 4 9 7 5 1 6 8 9 5 5 7 9 3 8 3 8 1 5 3
##  [2557] 5 0 5 5 3 8 6 7 7 7 3 7 0 5 9 0 2 5 5 3 1 7 7 8 6 5 9 3 8 9 5 3 7 9 1 7
##  [2593] 0 0 3 7 2 3 2 1 8 6 2 9 5 7 5 2 8 6 2 5 1 4 8 4 5 8 3 0 6 2 7 3 3 2 1 0
##  [2629] 7 3 4 0 3 9 3 2 8 9 0 3 8 0 7 6 5 4 7 3 0 0 8 6 2 5 1 1 0 0 4 4 0 1 2 3
##  [2665] 2 7 7 8 5 2 5 7 6 9 1 4 1 6 4 2 4 3 5 4 3 9 5 0 1 5 3 8 9 1 9 7 9 5 5 2
##  [2701] 7 4 6 0 1 1 1 0 4 4 7 6 3 0 0 4 3 0 6 1 9 6 1 3 8 1 2 5 6 2 7 3 6 0 1 9
##  [2737] 7 6 6 8 9 2 9 8 8 3 1 0 0 7 6 6 2 1 6 9 3 1 8 6 9 0 6 0 0 0 6 3 5 9 3 4
##  [2773] 5 5 8 5 3 0 4 0 2 9 6 8 2 3 1 2 1 1 5 6 9 8 0 6 6 5 5 3 8 6 2 1 4 5 4 3
##  [2809] 7 8 3 0 9 3 5 1 1 0 4 4 7 0 1 7 0 1 6 1 4 5 6 6 5 7 8 4 4 7 2 5 3 7 0 7
##  [2845] 7 9 6 4 2 8 5 7 8 3 9 5 8 9 9 8 6 2 8 4 2 3 6 1 1 8 9 3 4 0 7 9 6 7 1 4
##  [2881] 1 3 4 9 3 1 4 7 7 4 7 2 9 3 0 8 8 8 4 0 4 4 1 5 2 8 3 4 9 5 2 8 1 5 3 7
##  [2917] 9 4 2 5 6 2 5 9 3 5 9 2 1 9 5 3 0 6 9 8 4 0 4 7 2 9 0 1 0 3 1 6 5 8 1 5
##  [2953] 3 5 0 3 5 5 9 2 8 7 0 4 9 1 9 7 7 5 3 2 0 9 1 8 6 2 3 9 6 2 1 9 1 3 5 5
##  [2989] 0 3 8 3 3 7 6 6 0 1 4 0 6 9 8 1 2 9 9 5 9 7 3 7 8 0 1 3 0 4 6 1 0 2 5 8
##  [3025] 4 4 1 1 5 4 6 6 0 6 9 2 6 2 7 1 7 9 4 0 0 3 8 2 2 3 1 6 0 5 7 7 9 2 6 7
##  [3061] 7 7 8 6 8 8 4 6 8 4 1 2 8 2 3 9 4 0 3 7 3 2 3 3 7 3 4 0 6 2 0 8 1 5 3 5
##  [3097] 4 1 7 1 5 7 5 7 3 2 2 7 3 7 3 7 8 5 4 5 2 9 6 5 3 6 7 4 1 7 1 5 2 3 6 3
##  [3133] 1 4 2 6 7 4 3 8 0 6 2 1 6 5 3 9 1 9 3 2 1 8 4 4 6 5 8 6 9 7 7 8 6 9 7 3
##  [3169] 9 4 0 5 4 6 4 1 2 3 0 0 2 6 6 5 7 0 8 6 4 7 9 0 7 3 4 2 1 8 8 5 9 2 7 1
##  [3205] 8 8 8 2 7 6 0 1 2 7 1 0 8 3 6 0 5 3 6 2 8 7 0 1 4 2 1 1 4 4 4 4 7 1 6 2
##  [3241] 9 9 0 0 1 8 8 4 3 4 2 0 6 1 6 1 2 2 2 1 2 3 7 8 1 0 0 2 1 6 6 0 1 6 2 5
##  [3277] 1 7 4 8 2 1 4 3 8 3 9 9 4 8 3 4 7 2 7 5 7 0 4 3 3 2 6 7 6 0 0 6 7 7 0 5
##  [3313] 5 8 1 0 7 0 2 8 1 5 0 8 8 0 3 2 7 7 2 6 4 7 5 5 5 2 9 2 8 4 6 8 6 5 0 0
##  [3349] 8 7 6 1 7 1 1 2 7 4 0 0 7 7 6 3 8 6 4 2 0 9 4 0 5 7 8 2 7 4 7 1 1 3 6 6
##  [3385] 2 9 1 9 4 8 3 6 9 5 9 6 2 4 6 7 7 0 6 6 9 9 8 3 5 3 4 9 0 0 5 2 5 0 7 1
##  [3421] 1 1 0 7 6 7 9 6 6 4 1 4 3 1 1 2 2 4 1 0 8 7 6 3 4 0 0 6 3 3 0 7 1 7 1 1
##  [3457] 3 1 0 9 9 7 5 4 1 4 8 9 5 3 5 1 9 8 2 7 3 9 9 0 1 0 2 9 3 9 3 3 6 2 4 9
##  [3493] 8 3 7 4 0 4 7 8 4 9 8 1 9 7 5 9 2 8 2 2 0 2 2 3 8 4 6 8 4 8 2 4 6 7 9 3
##  [3529] 3 9 4 3 1 9 4 7 0 5 9 6 0 4 4 4 4 6 1 2 3 2 6 4 5 9 6 8 5 6 0 8 6 4 1 8
##  [3565] 6 5 2 8 4 5 5 4 7 7 0 7 8 2 2 3 7 0 1 8 0 7 1 9 8 7 5 5 9 1 7 5 4 3 1 2
##  [3601] 2 1 6 6 7 1 1 4 0 7 4 2 4 0 6 4 7 6 9 5 3 4 6 5 0 1 8 8 2 8 3 5 7 8 0 8
##  [3637] 5 7 1 1 0 1 3 7 8 5 0 7 1 1 0 1 1 4 5 2 7 6 2 3 0 2 8 5 9 6 9 7 2 1 3 6
##  [3673] 4 1 8 2 4 0 5 1 0 2 2 6 4 4 3 9 6 1 6 5 7 9 2 0 2 6 0 1 4 3 5 2 8 8 0 8
##  [3709] 8 9 0 9 6 7 6 3 9 3 4 7 7 7 4 9 0 6 4 4 4 2 7 2 8 1 0 0 7 8 3 3 3 1 3 7
##  [3745] 6 1 3 1 6 6 5 2 4 7 5 9 5 8 4 9 9 1 6 5 0 1 3 7 0 3 4 8 2 2 0 2 8 1 5 1
##  [3781] 6 8 8 9 1 2 1 3 5 1 0 9 4 4 8 3 2 5 9 7 6 6 2 0 0 0 5 8 8 1 5 2 3 8 5 1
##  [3817] 8 2 0 4 9 9 6 2 3 3 5 6 4 8 0 9 2 8 3 6 7 5 7 2 9 4 9 1 2 8 6 0 7 0 9 1
##  [3853] 1 5 7 5 9 9 1 9 5 9 2 5 0 4 1 0 8 9 0 8 9 8 9 4 2 5 7 9 8 9 8 0 9 9 6 8
##  [3889] 9 9 5 9 8 6 1 0 3 3 5 2 1 6 5 0 2 8 2 5 6 2 3 0 2 2 6 4 3 5 5 1 7 2 1 6
##  [3925] 9 1 9 9 5 5 1 6 2 2 8 6 7 1 4 6 0 6 0 5 3 2 2 3 6 8 9 8 5 3 8 5 4 5 2 0
##  [3961] 5 6 3 2 8 3 9 9 3 7 9 4 6 7 1 3 7 3 6 6 0 9 0 1 9 4 2 8 8 0 1 6 9 7 5 3
##  [3997] 4 7 4 9 9 4 3 6 3 1 1 7 6 9 1 8 4 1 1 9 9 4 3 6 8 1 6 0 4 1 3 7 7 4 9 5
##  [4033] 1 0 0 1 1 6 2 1 9 8 4 0 3 6 4 9 0 7 1 6 5 7 5 2 5 1 8 5 4 7 0 6 7 2 2 5
##  [4069] 8 1 0 4 5 7 1 3 5 1 9 0 0 6 0 7 3 1 8 3 9 7 0 0 8 9 5 9 8 3 2 7 2 9 7 2
##  [4105] 1 1 3 7 5 3 1 9 8 2 2 2 8 8 5 7 3 8 9 8 8 6 8 2 3 9 7 5 6 2 9 2 8 8 1 6
##  [4141] 2 8 7 9 1 8 0 1 7 2 0 7 5 1 9 0 2 0 9 8 6 2 3 0 3 8 0 2 1 1 1 1 4 2 9 7
##  [4177] 7 5 1 1 2 1 9 9 9 1 0 2 0 2 1 1 4 6 4 1 5 4 9 7 7 1 5 6 2 2 2 8 0 6 9 6
##  [4213] 1 9 7 7 1 4 8 5 3 4 3 4 7 7 5 0 7 4 8 8 1 5 3 9 5 9 7 6 9 0 3 6 3 9 8 2
##  [4249] 8 1 2 8 6 8 5 3 3 9 4 9 2 5 1 5 1 4 4 1 4 4 3 3 9 1 2 2 3 3 0 2 9 0 0 9
##  [4285] 9 6 0 9 3 7 8 4 1 9 9 7 2 7 9 9 5 9 5 1 1 8 3 5 1 9 5 3 5 4 9 5 9 3 1 9
##  [4321] 0 9 7 5 4 9 2 0 1 0 5 1 4 9 3 3 6 1 5 2 5 2 2 0 9 2 6 6 0 1 2 0 3 0 2 5
##  [4357] 5 7 9 5 3 0 8 9 5 0 3 2 5 9 0 8 8 4 5 8 8 4 5 4 8 5 4 9 2 2 1 2 6 8 8 7
##  [4393] 0 3 6 6 4 3 8 8 7 2 2 0 0 9 3 9 9 1 9 8 6 6 4 2 6 9 2 8 5 4 5 7 9 9 9 2
##  [4429] 1 8 3 4 0 7 8 3 9 3 4 6 5 6 2 3 9 2 6 0 0 6 1 2 8 7 9 8 2 0 4 7 7 5 0 5
##  [4465] 6 4 6 7 4 3 0 7 5 0 7 4 2 0 8 9 9 4 2 4 6 7 8 7 6 9 4 1 3 7 3 0 8 7 7 6
##  [4501] 9 3 9 2 2 9 2 1 8 3 2 9 6 8 4 0 1 2 8 4 5 2 7 8 1 1 3 0 3 5 7 0 3 1 9 3
##  [4537] 5 3 1 7 7 3 0 8 4 8 2 6 6 2 9 7 3 9 0 9 9 6 4 2 9 7 2 1 1 6 7 4 7 5 9 2
##  [4573] 8 2 1 4 4 5 7 6 1 3 2 5 9 9 3 6 1 1 4 6 9 7 2 1 5 1 4 6 3 8 1 1 0 3 1 6
##  [4609] 8 4 9 0 7 3 0 2 9 0 6 6 6 3 6 7 7 2 8 6 0 8 3 0 2 9 8 3 2 5 3 8 8 0 0 1
##  [4645] 9 5 1 3 9 6 0 1 4 1 7 1 2 3 7 9 7 4 9 9 3 9 2 8 2 7 1 8 0 9 1 0 1 7 7 9
##  [4681] 6 9 9 9 2 1 6 1 3 5 7 1 9 7 6 4 5 7 6 6 9 9 6 3 6 2 9 8 1 2 2 5 5 2 3 7
##  [4717] 2 1 0 1 0 4 5 2 8 2 8 3 5 1 7 7 1 1 2 9 7 8 4 0 3 0 7 8 8 4 7 7 8 5 8 4
##  [4753] 9 8 1 3 8 0 3 1 7 9 5 5 1 6 5 7 4 9 3 5 4 7 1 2 0 8 1 6 0 7 3 4 7 3 9 6
##  [4789] 0 8 6 4 8 7 7 9 3 8 6 9 7 2 3 4 0 2 1 3 3 5 5 7 2 4 6 7 2 8 3 0 8 7 8 4
##  [4825] 0 8 4 4 5 8 5 6 6 3 0 9 3 7 6 8 9 3 4 9 5 8 9 1 2 8 8 6 8 1 3 7 9 0 1 1
##  [4861] 9 7 0 8 1 7 4 5 7 1 2 1 1 3 9 6 2 1 2 8 8 7 6 6 9 3 7 0 5 2 8 0 5 4 3 8
##  [4897] 4 6 6 2 7 9 5 1 3 2 4 3 6 1 9 4 4 7 6 5 4 1 9 9 2 7 8 0 1 3 6 1 3 4 1 1
##  [4933] 1 5 6 0 7 0 7 2 3 2 5 2 2 9 4 9 8 1 2 1 6 1 2 7 4 0 0 0 8 2 2 9 2 2 7 9
##  [4969] 9 2 7 5 1 3 4 9 4 1 8 5 6 2 8 3 1 2 8 4 9 9 3 7 0 7 7 2 3 2 4 0 3 9 9 8
##  [5005] 4 1 0 6 0 9 6 8 6 1 1 9 8 9 2 3 5 5 9 4 2 1 9 4 3 9 6 0 4 0 6 0 1 2 3 4
##  [5041] 7 8 9 0 1 2 3 4 7 8 9 0 1 2 3 4 5 6 7 8 9 8 3 4 7 8 6 3 4 0 9 7 1 9 3 8
##  [5077] 4 7 3 0 9 1 4 5 4 6 2 0 6 2 1 1 1 1 7 2 4 7 5 2 9 4 5 8 4 2 9 7 0 0 7 5
##  [5113] 1 1 7 6 6 6 8 2 2 7 7 4 0 2 4 2 1 8 9 6 1 0 5 9 6 9 8 0 3 0 8 3 9 6 3 0
##  [5149] 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 5 4 8 7 4 7 7 3 9 8
##  [5185] 8 3 1 5 8 2 7 4 2 1 5 4 5 5 8 6 4 4 4 1 8 7 5 5 1 8 9 1 3 6 3 3 2 2 6 9
##  [5221] 9 6 5 5 3 3 8 1 6 5 6 8 1 9 7 6 8 3 7 4 7 0 9 0 0 3 7 9 3 0 2 0 1 0 1 0
##  [5257] 4 0 1 0 4 7 9 6 2 6 2 2 9 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
##  [5293] 2 3 4 5 6 7 8 9 8 0 5 6 6 0 8 0 2 3 7 9 4 7 1 9 1 7 1 4 0 0 4 1 7 5 7 1
##  [5329] 3 3 3 6 6 9 7 4 3 0 2 5 2 6 0 8 9 4 3 5 4 8 1 5 9 0 6 4 3 6 3 3 8 1 4 7
##  [5365] 5 7 2 2 0 0 1 7 7 9 5 9 8 9 6 8 8 2 3 6 1 2 9 8 9 5 2 6 2 4 8 4 6 5 0 1
##  [5401] 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7 4 2 0 9 0 1 5 8 8 0
##  [5437] 2 7 8 4 4 6 1 0 4 5 3 9 4 2 0 5 0 1 3 2 9 8 6 0 1 1 8 0 4 7 7 6 3 6 0 7
##  [5473] 3 5 4 2 4 1 8 3 5 6 7 0 6 7 1 2 5 8 1 9 3 8 2 8 7 6 7 1 4 6 2 9 3 0 1 2
##  [5509] 3 4 5 6 7 0 1 2 3 4 5 0 1 2 8 9 1 4 0 9 5 0 8 0 7 7 1 1 2 9 3 6 7 2 3 8
##  [5545] 1 2 9 8 8 7 1 7 1 1 0 3 4 2 6 4 7 4 2 7 4 9 1 0 6 8 5 5 5 3 5 9 7 4 8 5
##  [5581] 9 6 9 3 0 3 8 9 1 8 1 6 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2
##  [5617] 3 4 5 6 7 8 9 3 5 3 2 9 3 2 1 4 5 5 2 3 2 1 3 9 7 2 8 2 8 9 1 8 8 7 8 1
##  [5653] 0 0 7 7 8 7 5 0 6 1 5 7 4 6 1 2 5 0 7 9 9 0 3 8 2 4 8 1 8 6 5 9 0 0 0 3
##  [5689] 7 1 6 4 2 6 6 0 4 5 4 1 3 8 6 3 9 9 5 9 3 7 8 5 6 4 7 6 2 2 0 9 4 0 1 2
##  [5725] 3 4 5 6 7 8 9 0 1 2 7 5 6 0 1 2 3 4 5 6 8 7 1 3 2 8 0 7 5 9 9 6 0 9 4 1
##  [5761] 3 2 1 2 3 8 3 2 6 5 6 8 2 7 4 8 1 8 0 5 3 9 4 1 9 2 1 9 6 7 9 0 4 6 1 7
##  [5797] 3 8 7 2 9 6 5 8 3 9 0 5 7 1 6 1 0 9 3 3 4 4 0 6 2 5 4 2 3 4 6 0 0 2 0 1
##  [5833] 4 5 6 7 8 9 0 1 2 3 7 5 6 7 8 0 1 2 3 4 5 6 7 8 9 8 7 1 3 7 5 2 8 0 7 5
##  [5869] 9 9 0 9 1 1 5 8 8 6 3 2 1 8 3 2 6 5 6 0 4 1 0 5 3 1 9 2 1 9 6 0 4 6 1 7
##  [5905] 3 8 7 2 9 6 5 8 3 5 7 1 6 1 0 9 6 2 5 4 2 3 4 4 6 0 0 2 0 1 2 3 9 3 6 7
##  [5941] 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 8 4 5 6 7 8 9 8 6 5 0 6 8 9 4 1 9 3 8 0 4
##  [5977] 8 9 1 4 0 5 5 2 1 5 4 0 7 6 0 1 7 0 6 8 9 5 1 7 9 8 6 0 8 1 7 7 1 3 2 3
##  [6013] 1 4 2 0 0 7 8 4 6 4 9 3 8 4 7 2 3 6 3 6 9 6 3 2 2 4 6 9 0 2 5 5 1 3 3 9
##  [6049] 7 8 7 2 2 5 7 9 8 2 1 3 1 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
##  [6085] 2 3 4 5 6 7 8 9 1 2 6 5 3 0 7 0 4 1 4 3 6 7 2 3 1 2 1 2 9 6 0 1 3 0 2 7
##  [6121] 5 7 6 2 9 1 9 0 6 0 6 0 2 0 6 1 5 8 4 3 0 1 5 4 4 8 5 7 5 7 8 3 4 8 8 5
##  [6157] 2 9 7 1 3 8 1 0 7 5 9 6 9 4 7 7 9 9 3 4 4 3 8 6 2 0 1 2 3 4 5 6 7 8 9 0
##  [6193] 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 8 3 9 5 5 2 6 8 4 9 1 7 1 2 3 5
##  [6229] 9 6 9 1 1 1 2 9 5 6 8 1 2 0 7 7 5 8 2 9 8 9 0 4 6 7 1 3 4 5 6 0 3 6 8 7
##  [6265] 0 4 2 7 4 7 5 4 3 4 2 8 1 5 1 2 0 2 5 6 4 3 0 0 0 3 3 5 7 0 6 4 8 8 6 3
##  [6301] 4 6 9 9 8 2 7 7 1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7
##  [6337] 8 2 1 7 2 5 0 8 0 2 7 8 8 3 6 0 2 7 6 6 1 2 8 8 7 7 4 7 7 3 7 4 5 4 3 3
##  [6373] 8 4 1 1 9 7 4 3 7 3 3 0 2 5 5 6 6 3 5 2 5 9 9 8 4 1 0 6 0 9 6 8 8 5 6 1
##  [6409] 1 9 8 9 2 3 5 5 9 4 2 1 9 2 9 2 0 6 0 4 0 0 1 2 3 4 7 8 9 0 1 2 3 7 8 9
##  [6445] 0 1 2 3 4 7 8 9 7 3 0 3 1 8 7 6 4 0 2 6 8 3 2 8 1 2 0 7 1 0 4 4 5 8 0 6
##  [6481] 2 3 1 5 1 8 5 9 4 0 7 5 8 8 3 8 9 2 6 2 5 3 1 7 3 0 1 9 9 6 0 3 9 2 8 1
##  [6517] 4 3 5 2 9 2 5 8 9 5 0 1 2 4 5 6 0 1 2 3 4 5 6 7 1 2 3 4 5 1 0 4 5 6 6 3
##  [6553] 4 4 2 9 1 0 6 4 9 7 2 3 3 9 2 0 9 3 3 7 1 5 2 3 1 7 8 4 0 2 4 0 2 4 7 8
##  [6589] 0 7 0 6 9 3 2 8 6 7 5 7 5 1 0 8 1 6 7 2 9 7 9 5 8 6 2 6 2 8 1 7 5 0 1 1
##  [6625] 3 7 4 9 1 8 6 8 9 0 1 2 3 4 5 6 7 5 9 0 1 2 3 4 7 8 9 5 1 7 8 9 9 8 9 8
##  [6661] 4 1 7 7 3 3 7 6 6 6 1 9 0 1 7 6 3 2 1 7 1 3 9 1 7 6 8 4 1 4 3 6 9 6 1 4
##  [6697] 4 7 2 4 4 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 9 0 1 2 3 4 7 8 1 3 5 1 7 7
##  [6733] 2 1 4 8 3 4 4 3 9 7 4 1 2 3 5 9 1 6 0 1 0 0 2 7 7 1 1 4 0 4 7 3 6 8 0 3
##  [6769] 7 4 0 6 9 2 6 5 8 6 9 0 4 0 6 6 9 2 0 9 5 1 3 7 6 9 3 0 2 2 0 1 2 3 4 5
##  [6805] 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 2 1 7 2 5 0 8 0 2 7 8 8
##  [6841] 3 0 6 0 2 7 6 4 1 2 8 8 7 7 4 7 7 3 7 4 5 4 3 3 8 4 5 4 1 1 9 7 4 3 7 3
##  [6877] 3 0 2 5 5 6 3 1 5 2 5 9 9 8 4 1 0 6 0 9 6 8 8 5 6 1 1 9 8 9 2 3 5 5 9 4
##  [6913] 2 1 9 4 9 1 3 9 2 0 6 0 4 0 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
##  [6949] 1 2 3 4 5 6 7 8 9 3 8 0 7 1 0 7 5 5 6 9 0 1 0 0 8 3 4 3 1 5 0 0 9 5 3 4
##  [6985] 9 3 7 6 9 2 4 5 7 2 6 4 9 4 9 4 1 2 2 5 8 1 3 2 9 4 3 8 2 2 1 2 8 6 5 1
##  [7021] 6 7 2 1 3 9 3 8 7 5 7 0 7 4 8 8 5 0 6 6 3 7 6 9 9 4 8 4 1 0 6 6 0 1 2 3
##  [7057] 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7 4 0 4 0 1 7 9 5 1
##  [7093] 4 2 8 9 4 3 7 8 2 4 4 3 3 6 9 9 5 8 6 7 0 6 8 2 6 3 9 3 2 8 6 1 7 4 8 8
##  [7129] 9 0 3 3 9 0 5 2 9 4 1 0 3 7 5 8 7 7 8 2 9 7 1 2 6 4 2 5 2 3 6 6 5 0 0 2
##  [7165] 8 1 6 1 0 4 3 1 6 1 9 0 1 4 5 6 7 8 9 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9
##  [7201] 8 4 0 0 7 2 4 3 8 6 6 3 2 6 3 3 3 1 4 7 8 0 3 1 9 0 1 9 1 2 7 0 1 3 8 2
##  [7237] 9 2 7 6 5 5 9 9 8 2 9 1 3 2 3 4 3 1 9 0 9 3 6 8 7 0 1 0 5 8 2 7 7 0 1 2
##  [7273] 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 7 4 8 1 5 6 5 7
##  [7309] 2 8 6 3 3 8 6 5 4 0 9 1 7 2 9 1 5 1 3 2 2 3 0 6 4 3 7 6 9 0 4 8 1 4 0 6
##  [7345] 1 2 6 9 2 2 3 5 5 1 0 7 7 9 6 2 9 4 7 0 2 3 4 0 0 8 8 8 5 1 3 7 4 9 8 8
##  [7381] 9 0 9 8 9 0 2 6 5 6 7 4 7 5 4 1 3 5 3 1 2 3 4 5 6 1 2 3 4 6 0 1 2 4 5 6
##  [7417] 7 8 1 7 2 4 1 4 1 4 9 6 8 4 5 3 7 8 8 3 3 5 6 7 0 6 1 6 8 7 0 1 5 0 8 5
##  [7453] 0 1 5 8 4 2 3 9 7 6 9 1 9 0 6 7 1 2 3 9 2 4 5 5 3 7 5 3 1 8 2 2 3 0 2 9
##  [7489] 4 9 7 0 2 7 4 9 9 2 5 9 8 3 8 6 7 0 0 1 2 3 4 3 6 7 8 9 0 1 2 3 4 5 6 7
##  [7525] 8 9 0 1 2 3 4 5 6 7 8 9 0 0 7 2 6 5 5 3 7 8 6 6 6 6 4 3 8 8 3 0 1 9 0 5
##  [7561] 4 1 9 1 2 7 0 1 3 8 2 9 2 7 4 2 6 5 5 9 9 1 1 5 7 6 8 2 9 4 3 1 9 0 9 3
##  [7597] 6 8 7 0 1 0 5 8 2 7 7 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 8 9 0 1 2 3 4 5 6
##  [7633] 7 8 9 2 1 2 1 3 9 9 8 5 3 7 0 7 7 5 7 9 9 4 7 0 3 4 1 5 8 1 4 8 4 1 8 6
##  [7669] 6 4 6 0 5 5 3 3 5 7 2 5 9 6 9 2 6 2 1 2 0 8 3 8 3 0 8 7 4 9 5 0 9 7 0 0
##  [7705] 4 6 0 9 1 6 2 7 6 8 3 5 2 1 8 3 8 6 1 0 2 1 4 0 1 2 3 4 5 6 7 8 9 0 1 2
##  [7741] 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7 6 4 7 6 2 3 4 8 7 8 6 9 8 3 2 2 8 4
##  [7777] 8 5 6 5 0 2 0 1 1 2 9 6 8 2 1 0 6 5 2 9 7 5 3 9 2 7 1 8 3 8 1 9 5 5 0 1
##  [7813] 8 9 8 2 6 0 4 5 0 2 1 8 6 7 5 9 9 3 0 3 1 4 4 0 4 9 0 1 2 3 5 6 7 8 0 1
##  [7849] 2 3 5 6 7 8 9 0 1 2 3 5 6 7 8 9 9 7 0 9 0 1 5 8 8 0 9 3 2 7 8 4 6 1 0 4
##  [7885] 9 4 2 0 5 0 1 6 9 3 2 9 1 6 0 8 1 8 7 7 6 3 6 0 7 2 4 1 7 0 6 7 1 2 5 8
##  [7921] 1 2 2 8 7 6 8 7 1 6 2 9 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2
##  [7957] 3 4 5 6 7 8 9 8 9 5 7 0 3 1 6 8 4 1 5 6 4 2 7 8 1 3 4 3 4 7 2 0 5 0 8 9
##  [7993] 2 3 2 3 5 5 7 8 4 9 9 7 1 1 9 0 7 8 3 4 8 6 3 8 0 9 6 2 1 0 1 0 6 2 3 8
##  [8029] 9 0 7 2 3 4 5 5 2 8 5 4 6 6 6 7 9 1 8 2 1 5 3 4 7 9 4 0 0 0 1 2 3 4 5 6
##  [8065] 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 9 0 1 3 1 5 1 2 4 9 8 4 6 8 0 1
##  [8101] 1 9 2 6 6 8 7 4 2 9 7 0 2 1 0 3 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8
##  [8137] 9 0 1 2 3 4 5 6 7 8 9 8 6 5 9 7 0 2 3 4 3 8 5 1 5 2 3 0 1 2 1 3 2 6 5 3
##  [8173] 0 7 2 7 4 6 4 0 5 9 9 8 9 5 3 1 7 4 7 6 5 4 0 0 6 6 2 0 6 3 7 7 4 4 3 9
##  [8209] 2 8 9 6 0 9 5 3 8 8 7 1 4 0 4 8 5 2 3 9 0 1 9 1 5 1 7 4 8 6 2 1 6 8 8 0
##  [8245] 1 2 9 4 7 8 9 0 1 2 3 4 6 7 8 9 0 1 2 3 4 7 8 9 1 4 5 3 3 0 9 5 4 9 0 8
##  [8281] 4 6 7 0 7 7 1 6 9 1 3 6 2 3 8 2 3 8 9 5 8 8 7 1 7 1 1 0 3 4 2 4 4 7 4 2
##  [8317] 7 4 2 9 2 7 9 2 1 6 6 5 3 4 8 5 9 6 9 0 6 3 0 8 1 6 0 0 1 2 3 4 5 6 7 0
##  [8353] 1 2 3 4 7 8 9 0 1 2 3 4 7 2 5 1 6 4 3 9 9 0 9 7 1 6 4 3 6 2 0 9 8 6 5 7
##  [8389] 0 0 1 7 4 3 2 4 1 3 7 6 4 7 7 7 9 8 4 3 6 2 8 3 5 8 0 5 4 7 1 3 1 7 9 6
##  [8425] 2 0 9 1 7 3 3 9 1 6 4 3 9 8 2 1 8 6 4 1 5 5 6 5 0 1 2 3 4 5 6 7 8 9 0 1
##  [8461] 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 6 9 7 0 2 3 4 3 8 5 1 3 0 1 2 1 3 2
##  [8497] 0 7 2 6 4 0 5 9 9 8 9 5 3 1 7 4 7 0 0 6 6 6 3 7 9 2 8 9 8 7 1 9 0 4 8 5
##  [8533] 2 3 9 0 1 9 1 5 1 7 6 1 2 1 6 8 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 0
##  [8569] 1 2 3 5 6 7 8 1 0 4 5 6 6 3 4 4 2 8 1 0 6 4 9 7 2 9 2 0 9 3 3 9 1 5 2 3
##  [8605] 1 6 7 3 7 8 4 0 2 4 0 2 4 7 8 0 7 0 6 9 3 2 4 8 6 0 5 7 5 1 0 8 1 6 7 2
##  [8641] 9 7 9 5 6 5 2 6 2 8 1 7 5 5 7 3 5 0 1 1 3 8 4 9 4 5 1 8 6 8 9 0 1 2 3 4
##  [8677] 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 3 5 3 2 9 3 2 1 4 5 5
##  [8713] 2 3 2 1 3 9 7 2 1 2 8 9 1 8 8 7 8 1 0 0 6 7 7 8 7 5 0 6 1 5 7 4 6 1 2 5
##  [8749] 0 7 9 9 0 3 4 4 8 4 1 8 6 5 9 0 0 0 3 7 1 6 4 6 0 4 5 4 1 3 8 6 3 9 9 5
##  [8785] 9 3 7 8 5 6 4 7 6 2 2 0 9 4 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
##  [8821] 2 3 4 5 6 7 8 9 6 4 2 6 4 7 5 5 4 7 2 9 3 9 3 8 2 0 9 5 6 0 1 0 6 5 3 5
##  [8857] 3 8 0 0 3 4 1 5 3 0 8 3 0 6 2 7 8 1 7 1 3 8 5 4 2 0 9 7 6 7 4 1 6 2 6 7
##  [8893] 1 9 8 0 6 9 4 9 9 6 2 3 7 1 9 2 2 5 3 7 8 0 1 2 3 4 7 8 9 0 1 2 3 4 7 8
##  [8929] 9 0 1 7 8 9 8 9 2 6 1 3 5 4 8 2 6 4 3 4 5 9 2 0 3 9 4 9 7 3 8 7 4 4 9 8
##  [8965] 5 8 2 6 6 2 3 1 3 2 7 3 1 9 0 1 1 3 5 0 7 8 1 5 1 4 6 0 0 4 9 1 6 6 9 0
##  [9001] 7 6 1 1 0 1 2 3 4 2 2 3 4 5 6 2 0 1 2 2 8 6 3 9 2 1 9 3 9 6 1 7 2 4 4 5
##  [9037] 7 0 0 1 6 6 8 2 7 7 2 4 2 1 6 1 0 6 9 8 3 9 6 3 0 1 2 3 4 5 6 7 8 9 0 1
##  [9073] 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 6 8 9 9 0 1 2 4 4 3 7 4 4 4 0 3 8
##  [9109] 7 5 8 2 1 7 5 3 8 5 2 5 1 1 6 2 1 3 8 6 4 2 6 2 5 5 0 2 8 0 6 8 1 7 9 1
##  [9145] 9 2 6 7 6 6 8 7 4 9 2 1 3 3 0 5 5 8 0 3 7 9 7 0 2 7 9 1 7 8 0 3 5 3 6 0
##  [9181] 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 7 8 9 6 4 2 6 4 7 8 9 2
##  [9217] 9 3 9 3 0 0 1 0 4 2 6 3 5 3 0 3 4 1 5 3 0 8 3 0 6 1 7 8 0 9 2 6 7 1 9 6
##  [9253] 9 4 9 9 6 7 1 2 5 3 7 8 0 1 2 4 5 6 7 8 9 0 1 3 4 5 6 7 5 0 1 3 4 7 8 9
##  [9289] 7 5 5 1 9 9 7 1 0 0 5 9 7 1 7 2 2 3 6 8 3 2 0 0 6 1 7 5 8 6 2 9 4 8 8 7
##  [9325] 1 0 8 7 7 5 8 5 3 4 6 1 1 5 5 0 7 2 3 6 4 1 2 4 1 5 4 2 0 4 8 6 1 9 0 2
##  [9361] 5 6 9 3 6 3 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 5 6 7 8 1
##  [9397] 0 9 5 7 5 1 8 6 9 0 4 1 9 3 8 4 4 7 0 1 9 2 8 7 8 2 5 9 6 0 6 5 5 3 3 3
##  [9433] 9 8 1 1 0 6 1 0 0 6 2 1 1 3 2 7 7 8 8 7 8 4 6 0 2 0 7 0 3 6 8 7 1 5 9 9
##  [9469] 3 7 2 4 9 4 3 6 2 2 5 3 2 5 5 9 4 1 7 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
##  [9505] 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 0 1 2 7 5 3 4 4 0 0 6 9 6 6 5 7 2 3 4 4 9
##  [9541] 1 4 0 7 9 5 7 2 3 1 4 4 0 9 9 6 1 8 3 3 7 3 9 8 8 4 7 7 6 2 1 9 8 7 8 8
##  [9577] 7 2 2 3 9 3 3 5 5 0 7 4 5 6 5 1 4 1 1 2 8 2 6 1 5 0 1 2 3 4 5 6 7 8 9 0
##  [9613] 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 8 0 6 0 2 2 3 7 9 4 7 1 9 1 7 1 4 0
##  [9649] 0 1 7 5 7 1 3 3 3 1 6 9 7 1 3 0 7 6 0 8 9 7 3 5 4 8 1 5 9 0 6 3 3 8 1 4
##  [9685] 7 5 2 0 0 1 7 8 9 6 8 8 2 3 6 1 2 9 5 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
##  [9721] 6 7 8 9 0 1 2 3 4 6 6 7 8 9 7 4 6 1 4 0 9 9 3 7 8 0 7 5 8 5 3 2 2 0 5 8
##  [9757] 6 0 3 8 1 0 3 0 4 7 4 9 0 9 0 7 1 7 1 6 6 0 6 2 8 7 6 4 9 9 5 3 7 4 3 0
##  [9793] 9 6 6 1 1 3 2 1 0 0 1 2 3 4 7 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 7 8 9 0 8
##  [9829] 3 9 5 5 2 6 8 4 1 7 1 7 3 5 6 9 1 1 1 2 1 2 0 7 7 5 8 2 9 8 6 7 3 4 6 8
##  [9865] 7 0 4 2 7 7 5 4 3 4 2 8 1 5 1 0 2 3 3 5 7 0 6 8 6 3 9 9 8 2 7 7 1 0 1 7
##  [9901] 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 7 8 9 7 8 6 4 1 9 3 8 4 4 7 0 1 9 2 8 7
##  [9937] 8 2 6 0 6 5 3 3 8 9 1 4 0 6 1 0 0 6 2 1 1 7 7 8 4 6 0 7 0 3 6 8 7 1 5 2
##  [9973] 4 9 4 3 6 4 1 7 2 6 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6

g_test

##     [1] 7 2 1 0 4 1 4 9 5 9 0 6 9 0 1 5 9 7 3 4 9 6 6 5 4 0 7 4 0 1 3 1 3 4 7 2
##    [37] 7 1 2 1 1 7 4 2 3 5 1 2 4 4 6 3 5 5 6 0 4 1 9 5 7 8 9 3 7 4 6 4 3 0 7 0
##    [73] 2 9 1 7 3 2 9 7 7 6 2 7 8 4 7 3 6 1 3 6 9 3 1 4 1 7 6 9 6 0 5 4 9 9 2 1
##   [109] 9 4 8 7 3 9 7 4 4 4 9 2 5 4 7 6 7 9 0 5 8 5 6 6 5 7 8 1 0 1 6 4 6 7 3 1
##   [145] 7 1 8 2 0 2 9 9 5 5 1 5 6 0 3 4 4 6 5 4 6 5 4 5 1 4 4 7 2 3 2 7 1 8 1 8
##   [181] 1 8 5 0 8 9 2 5 0 1 1 1 0 9 0 3 1 6 4 2 3 6 1 1 1 3 9 5 2 9 4 5 9 3 9 0
##   [217] 3 6 5 5 7 2 2 7 1 2 8 4 1 7 3 3 8 8 7 9 2 2 4 1 5 9 8 7 2 3 0 4 4 2 4 1
##   [253] 9 5 7 7 2 8 2 6 8 5 7 7 9 1 8 1 8 0 3 0 1 9 9 4 1 8 2 1 2 9 7 5 9 2 6 4
##   [289] 1 5 8 2 9 2 0 4 0 0 2 8 4 7 1 2 4 0 2 7 4 3 3 0 0 3 1 9 6 5 2 5 9 2 9 3
##   [325] 0 4 2 0 7 1 1 2 1 5 3 3 9 7 8 6 5 6 1 3 8 1 0 5 1 3 1 5 5 6 1 8 5 1 7 9
##   [361] 4 6 2 2 5 0 6 5 6 3 7 2 0 8 8 5 4 1 1 4 0 3 3 7 6 1 6 2 1 9 2 8 6 1 9 5
##   [397] 2 5 4 4 2 8 3 8 2 4 5 0 3 1 7 7 5 7 9 7 1 9 2 1 4 2 9 2 0 4 9 1 4 8 1 8
##   [433] 4 5 9 8 8 3 7 6 0 0 3 0 2 6 6 4 9 3 3 3 2 3 9 1 2 6 8 0 5 6 6 6 3 8 8 2
##   [469] 7 5 8 9 6 1 8 4 1 2 5 9 1 9 7 5 4 0 8 9 9 1 0 5 2 3 7 8 9 4 0 6 3 9 5 2
##   [505] 1 3 1 3 6 5 7 4 2 2 6 3 2 6 5 4 8 9 7 1 3 0 3 8 3 1 9 3 4 4 6 4 2 1 8 2
##   [541] 5 4 8 8 4 0 0 2 3 2 7 7 0 8 7 4 4 7 9 6 9 0 9 8 0 4 6 0 6 3 5 4 8 3 3 9
##   [577] 3 3 3 7 8 0 8 2 1 7 0 6 5 4 3 8 0 9 6 3 8 0 9 9 6 8 6 8 5 7 8 6 0 2 4 0
##   [613] 2 2 3 1 9 7 5 1 0 8 4 6 2 6 7 9 3 2 9 8 2 2 9 2 7 3 5 9 1 8 0 2 0 5 2 1
##   [649] 3 7 6 7 1 2 5 8 0 3 7 2 4 0 9 1 8 6 7 7 4 3 4 9 1 9 5 1 7 3 9 7 6 9 1 3
##   [685] 7 8 3 3 6 7 2 8 5 8 5 1 1 4 4 3 1 0 7 7 0 7 9 4 4 8 5 5 4 0 8 2 1 0 8 4
##   [721] 5 0 4 0 6 1 7 3 2 6 7 2 6 9 3 1 4 6 2 5 4 2 0 6 2 1 7 3 4 1 0 5 4 3 1 1
##   [757] 7 4 9 9 4 8 4 0 2 4 5 1 1 6 4 7 1 9 4 2 4 1 5 5 3 8 3 1 4 5 6 8 9 4 1 5
##   [793] 3 8 0 3 2 5 1 2 8 3 4 4 0 8 8 3 3 1 7 3 5 9 6 3 2 6 1 3 6 0 7 2 1 7 1 4
##   [829] 2 4 2 1 7 9 6 1 1 2 4 8 1 7 7 4 8 0 7 3 1 3 1 0 7 7 0 3 5 5 2 7 6 6 9 2
##   [865] 8 3 5 2 2 5 6 0 8 2 9 2 8 8 8 8 7 4 9 3 0 6 6 3 2 1 3 2 2 9 3 0 0 5 7 8
##   [901] 1 4 4 6 0 2 9 1 4 7 4 7 3 9 8 8 4 7 1 2 1 2 2 3 2 3 2 3 9 1 7 4 0 3 5 5
##   [937] 8 6 3 2 6 7 6 6 3 2 7 8 1 1 7 5 6 4 9 5 1 3 3 4 7 8 9 1 1 6 9 1 4 4 5 4
##   [973] 0 6 2 2 3 1 5 1 2 0 3 8 1 2 6 7 1 6 2 3 9 0 1 2 2 0 8 9 9 0 2 5 1 9 7 8
##  [1009] 1 0 4 1 7 9 6 4 2 6 8 1 3 7 5 4 4 1 8 1 3 8 1 2 5 8 0 6 2 1 1 7 1 5 3 4
##  [1045] 6 9 5 0 9 2 2 4 8 2 1 7 2 4 9 4 4 0 3 9 2 2 3 3 8 3 5 7 3 5 8 1 2 4 4 6
##  [1081] 4 9 5 1 0 6 9 5 9 5 9 7 3 8 0 3 7 1 3 6 7 8 5 9 7 9 6 9 6 3 7 4 4 5 3 5
##  [1117] 4 7 8 7 8 0 7 6 8 8 7 3 3 1 9 5 2 7 3 5 1 1 2 1 4 7 4 7 5 4 5 4 0 8 3 6
##  [1153] 9 6 0 2 7 4 4 4 4 6 6 4 7 9 3 4 5 5 8 7 3 7 2 7 0 2 4 1 1 6 6 9 2 8 7 2
##  [1189] 0 1 5 0 9 1 7 0 6 0 8 6 8 1 8 0 3 3 7 2 3 6 2 1 6 1 1 3 7 9 0 8 0 5 4 0
##  [1225] 2 8 7 2 9 8 4 0 9 5 8 5 1 2 1 3 1 7 4 5 7 2 0 9 8 8 6 2 5 4 1 9 2 1 5 8
##  [1261] 7 0 2 4 4 3 6 8 8 2 4 0 5 0 4 4 7 9 3 4 1 5 9 7 3 5 8 8 0 5 3 3 6 6 0 1
##  [1297] 6 0 3 5 4 4 1 2 9 1 4 6 9 9 3 9 8 4 4 3 1 3 1 8 8 7 9 4 8 8 7 9 7 1 4 5
##  [1333] 6 0 5 2 2 2 1 5 5 2 4 9 6 2 7 7 2 2 1 1 2 8 3 7 2 4 1 7 1 7 6 7 8 2 7 3
##  [1369] 1 7 5 8 2 6 2 2 5 6 5 0 9 2 4 3 3 9 7 6 6 8 0 4 1 5 8 2 9 1 8 0 6 7 2 1
##  [1405] 0 5 5 2 0 2 2 0 2 4 9 8 0 9 9 4 6 5 4 9 1 8 3 4 9 9 1 2 2 8 1 9 6 4 0 9
##  [1441] 4 8 3 8 6 0 2 5 1 9 6 2 9 4 0 9 6 0 6 2 5 4 2 3 8 4 5 5 0 3 8 5 3 5 8 6
##  [1477] 5 7 6 3 3 9 6 1 1 2 9 0 4 3 3 6 9 5 7 3 7 7 7 8 7 9 8 3 0 7 2 7 9 4 5 4
##  [1513] 9 3 2 1 4 0 2 3 7 5 7 8 8 5 0 1 1 4 8 3 9 0 0 0 6 6 2 3 7 8 4 7 7 9 2 4
##  [1549] 1 4 5 2 4 9 9 1 8 4 0 9 8 4 8 7 7 0 7 8 8 6 0 4 8 8 2 4 7 6 6 6 4 7 1 8
##  [1585] 8 2 3 6 3 0 0 3 7 6 9 7 9 9 5 4 3 3 6 1 2 3 7 3 3 2 0 3 3 8 4 3 6 3 5 0
##  [1621] 2 0 9 0 7 4 6 9 3 5 1 9 6 1 4 5 4 5 0 5 9 5 2 1 2 9 1 9 9 4 0 8 4 5 2 9
##  [1657] 2 1 2 1 7 3 6 8 8 4 9 1 9 8 5 7 5 1 1 8 6 5 2 4 4 3 2 3 5 6 8 8 6 2 3 1
##  [1693] 0 5 8 9 2 9 6 7 0 4 8 7 1 7 4 1 0 9 7 2 0 0 9 1 7 8 7 8 4 7 2 0 4 6 0 3
##  [1729] 1 1 3 3 9 6 7 4 1 5 3 0 8 7 3 9 6 9 3 5 0 2 7 4 5 1 7 5 8 0 8 8 1 5 0 3
##  [1765] 0 3 1 4 0 3 7 2 7 1 8 0 7 0 4 3 1 9 8 7 7 1 4 9 9 3 2 1 7 9 0 2 0 3 3 7
##  [1801] 6 9 2 3 3 7 7 0 0 7 5 2 9 8 7 4 4 2 6 6 1 9 6 8 2 9 0 8 3 1 1 6 3 5 1 1
##  [1837] 1 3 1 2 3 0 2 0 1 3 5 5 7 4 8 9 6 9 6 8 3 6 6 8 5 1 4 2 4 4 5 1 1 9 0 2
##  [1873] 4 9 5 7 1 8 8 5 6 9 8 7 1 1 6 7 6 3 2 2 0 8 9 2 5 1 0 8 1 9 5 7 9 6 9 0
##  [1909] 6 1 5 5 8 3 8 2 6 5 0 7 4 6 1 3 4 7 3 2 3 4 2 5 2 7 1 7 2 6 4 1 5 7 8 6
##  [1945] 0 1 8 2 5 7 7 6 9 3 5 8 4 2 4 0 8 8 3 4 9 2 7 5 8 6 5 6 0 8 6 7 3 6 4 9
##  [1981] 4 6 6 3 2 4 1 0 1 4 6 2 9 1 1 0 6 3 9 5 6 5 6 5 8 4 6 4 3 9 1 3 4 1 9 1
##  [2017] 7 1 1 9 3 5 4 0 7 3 6 1 7 5 5 3 3 0 1 5 7 5 8 6 5 1 0 4 2 3 4 6 7 9 8 1
##  [2053] 8 4 9 2 8 6 2 7 0 0 6 7 5 8 6 0 9 3 7 1 3 5 4 3 3 5 5 6 3 0 2 3 4 2 3 0
##  [2089] 9 9 4 7 2 8 4 7 0 6 2 8 5 2 8 5 7 3 0 8 2 3 2 8 2 5 5 7 6 4 6 8 4 8 2 7
##  [2125] 4 5 2 0 3 9 4 6 7 2 5 6 1 1 2 3 6 7 8 7 6 4 8 9 4 8 6 3 8 3 1 0 6 2 2 5
##  [2161] 6 9 5 8 1 4 1 7 8 4 6 1 8 4 3 1 2 8 0 8 5 9 1 4 2 0 2 7 0 9 0 2 5 7 6 7
##  [2197] 9 4 2 6 2 4 4 8 0 4 4 5 8 0 6 8 9 8 5 6 9 0 4 8 7 1 3 4 5 8 0 9 1 3 3 6
##  [2233] 9 8 7 1 0 5 7 1 7 5 2 7 9 1 8 5 2 4 9 4 7 2 2 3 4 9 1 9 2 1 7 9 4 4 1 6
##  [2269] 7 2 7 8 8 1 9 7 1 1 7 5 3 3 5 1 3 7 6 1 3 8 7 5 9 9 0 0 2 8 8 2 3 7 1 3
##  [2305] 0 3 4 4 3 8 9 2 3 9 7 1 1 7 0 4 9 6 5 9 1 7 0 2 0 0 4 6 7 0 7 1 4 6 4 5
##  [2341] 4 9 9 1 7 9 5 3 3 8 2 3 6 2 2 1 1 1 1 1 6 9 8 4 3 7 1 6 4 5 0 4 7 4 2 4
##  [2377] 0 7 0 1 9 8 8 6 0 0 4 9 6 8 2 2 3 8 4 8 2 2 1 7 5 4 4 0 4 3 9 7 3 1 0 1
##  [2413] 2 5 9 2 1 0 1 8 9 1 6 8 3 8 9 3 6 2 8 3 2 2 1 0 4 2 9 2 4 3 7 9 1 5 2 4
##  [2449] 9 0 3 8 5 3 6 0 9 4 6 2 5 0 2 7 4 6 6 8 6 6 8 6 9 1 7 2 5 9 9 0 7 2 7 6
##  [2485] 7 0 6 5 2 4 7 2 0 9 9 2 2 9 4 4 2 3 3 2 1 7 0 7 6 4 1 3 8 7 4 5 9 2 5 1
##  [2521] 8 7 3 7 1 5 5 0 9 1 4 0 6 3 3 6 0 4 9 7 5 1 6 8 9 5 5 7 9 3 8 3 8 1 5 3
##  [2557] 5 0 5 5 3 8 6 7 7 7 3 7 0 5 9 0 2 5 5 3 1 7 7 8 6 5 9 3 8 9 5 3 7 9 1 7
##  [2593] 0 0 3 7 2 5 8 1 8 6 2 9 5 7 5 7 8 6 2 5 1 4 8 4 5 8 3 0 6 2 7 3 3 2 1 0
##  [2629] 7 3 4 0 3 9 3 2 8 9 0 3 8 0 7 6 5 4 7 3 9 0 8 6 2 5 6 1 0 0 4 4 0 1 2 3
##  [2665] 2 7 7 8 5 2 5 7 6 9 1 4 1 6 4 2 4 3 5 4 3 9 5 0 1 5 3 8 9 1 9 7 9 5 5 2
##  [2701] 7 4 6 0 1 1 1 0 4 4 7 6 3 0 0 4 3 0 6 1 9 6 1 3 8 1 2 5 6 2 7 3 6 0 1 9
##  [2737] 7 6 6 8 9 2 9 5 8 3 1 0 0 7 6 6 2 1 6 9 3 1 8 6 9 0 6 0 0 0 6 3 5 9 3 4
##  [2773] 5 5 8 5 3 0 4 0 2 9 6 8 2 3 1 2 1 1 5 6 9 8 0 6 6 5 5 3 8 6 2 1 4 5 4 3
##  [2809] 7 8 5 0 9 3 5 1 1 0 4 4 7 0 1 7 0 1 6 1 4 5 6 6 5 7 8 4 4 7 2 5 3 7 0 7
##  [2845] 7 9 6 4 2 8 5 7 8 3 9 5 8 9 9 8 6 2 8 9 2 3 6 1 1 8 9 3 4 0 7 9 6 4 1 4
##  [2881] 1 3 4 9 3 1 4 7 7 4 7 2 9 3 0 8 8 8 4 0 4 4 1 5 2 8 3 4 9 5 2 8 1 5 3 7
##  [2917] 9 4 2 5 6 3 5 9 3 5 9 3 1 9 5 3 0 6 9 8 4 0 4 9 2 9 0 1 0 3 1 6 5 8 1 5
##  [2953] 3 3 0 3 5 5 9 2 8 7 0 4 9 1 9 7 7 5 5 2 0 9 1 8 6 2 3 9 6 2 1 9 1 3 5 5
##  [2989] 0 3 8 3 3 7 6 6 0 1 4 0 6 9 8 1 2 9 9 5 9 7 3 7 8 0 1 3 0 4 6 1 0 2 5 8
##  [3025] 4 4 1 1 5 4 6 6 0 6 9 2 6 2 7 1 7 9 4 0 0 3 8 2 2 3 1 6 0 5 7 7 9 2 6 7
##  [3061] 9 7 8 6 8 8 4 6 8 4 1 2 8 1 3 9 4 0 3 7 3 2 3 3 7 3 4 0 6 2 0 8 1 5 3 5
##  [3097] 4 1 7 1 5 7 5 7 3 2 2 7 3 7 3 7 8 5 4 5 2 5 6 5 3 6 7 4 1 7 1 5 2 3 6 3
##  [3133] 1 4 2 6 7 4 3 8 0 6 2 1 6 5 3 9 1 9 3 2 1 8 4 4 6 5 8 6 9 7 7 8 6 9 7 3
##  [3169] 9 4 0 5 4 6 4 1 2 3 0 0 2 6 6 5 7 0 8 6 4 7 9 0 7 3 4 2 1 8 8 5 9 2 7 1
##  [3205] 8 8 8 2 7 6 0 1 2 7 1 0 8 3 6 0 5 3 6 2 8 7 0 1 4 2 1 1 4 4 4 4 7 1 6 2
##  [3241] 9 9 0 0 1 8 8 4 3 4 2 0 6 1 6 1 2 2 2 1 2 3 7 8 1 0 0 2 1 6 6 0 1 6 2 5
##  [3277] 1 7 4 8 2 1 4 3 8 3 9 9 4 8 3 4 7 2 7 5 7 0 4 3 3 2 6 7 6 0 0 6 7 7 0 5
##  [3313] 5 8 1 0 7 0 2 8 1 5 0 8 8 0 3 2 7 7 2 6 4 7 5 5 5 2 9 2 8 4 6 8 6 5 0 0
##  [3349] 8 7 6 1 7 1 1 2 7 4 0 0 7 7 6 3 8 6 4 2 0 9 4 0 5 7 8 2 7 4 7 1 1 3 6 6
##  [3385] 2 9 1 9 4 8 3 6 9 5 9 6 2 4 6 7 7 0 6 6 9 4 8 3 5 3 4 9 0 0 5 2 5 0 7 1
##  [3421] 1 1 6 7 6 7 9 6 6 4 1 4 3 1 1 2 2 4 1 0 8 7 6 3 4 0 0 6 3 3 0 7 1 7 1 1
##  [3457] 3 1 0 9 9 7 5 4 1 4 8 9 5 3 5 1 9 8 2 3 3 9 9 0 1 0 2 9 3 9 3 3 6 2 4 9
##  [3493] 8 3 7 4 0 4 7 8 4 9 8 9 9 7 5 9 2 8 2 2 0 2 2 3 8 4 6 8 6 8 2 4 6 7 9 3
##  [3529] 3 9 4 3 1 4 4 7 0 5 9 6 0 4 4 4 4 6 1 2 3 3 6 4 5 9 6 8 5 6 5 8 6 4 1 8
##  [3565] 6 5 2 8 4 5 5 4 7 7 0 7 8 2 2 3 7 0 1 8 0 7 1 9 8 7 5 5 9 1 7 5 4 9 1 2
##  [3601] 2 1 6 6 7 1 1 4 0 7 4 2 4 0 6 4 7 6 9 5 3 4 6 5 0 1 8 8 2 8 3 5 7 8 0 8
##  [3637] 5 7 1 1 0 1 3 7 8 5 0 7 1 1 0 1 1 4 5 2 7 6 2 3 0 2 8 5 9 6 9 7 2 1 3 6
##  [3673] 4 1 8 2 4 0 5 1 0 2 2 6 4 4 3 9 6 1 6 5 7 9 2 0 2 6 0 1 4 3 5 2 8 8 0 8
##  [3709] 8 9 0 9 6 7 6 3 9 3 4 7 7 7 4 9 0 6 4 8 4 2 7 2 8 1 0 0 7 8 3 3 3 1 3 7
##  [3745] 6 1 3 1 6 6 5 7 4 7 5 9 5 8 4 9 9 1 6 5 0 1 3 7 0 3 4 8 2 2 0 2 5 1 5 1
##  [3781] 4 8 8 9 1 2 1 3 5 1 0 9 4 4 8 3 2 5 9 7 6 6 2 0 0 0 5 8 7 1 5 2 3 8 5 1
##  [3817] 8 2 0 4 9 9 6 2 3 3 5 6 4 8 0 9 2 8 3 6 7 5 7 2 9 4 9 1 2 8 6 0 7 0 9 1
##  [3853] 1 6 7 5 9 9 1 9 5 9 2 5 0 4 1 0 8 9 0 8 9 8 9 4 2 5 7 9 8 9 8 0 9 9 6 8
##  [3889] 9 9 5 9 8 5 1 0 3 3 5 2 1 6 5 0 2 8 1 5 6 2 3 0 2 2 6 4 3 5 5 1 7 2 1 6
##  [3925] 9 1 9 9 5 5 1 6 2 2 8 6 7 1 4 6 0 4 0 3 3 2 2 3 6 8 9 8 5 3 8 5 4 5 2 0
##  [3961] 5 6 3 2 8 3 9 9 5 7 9 4 6 7 1 3 7 3 6 6 0 9 0 1 9 9 2 8 8 0 1 6 9 7 5 3
##  [3997] 4 7 4 9 9 4 3 6 3 1 1 7 6 9 1 8 4 1 1 9 9 4 3 6 8 1 6 0 4 1 3 7 7 4 9 5
##  [4033] 1 0 0 1 1 6 2 1 9 8 4 0 3 6 4 9 0 7 1 6 5 7 5 2 5 1 8 5 4 7 0 6 7 0 2 5
##  [4069] 8 1 0 4 5 7 1 8 5 1 9 0 0 6 0 7 3 1 8 3 9 7 0 0 8 9 5 9 8 3 2 7 2 9 7 2
##  [4105] 1 1 3 7 5 3 1 9 8 2 2 2 8 8 5 7 3 8 9 8 8 6 8 2 3 9 7 5 6 2 9 2 8 8 1 6
##  [4141] 8 8 7 9 1 8 0 1 7 2 0 7 5 1 9 0 2 0 9 8 6 2 3 9 3 8 0 2 1 1 1 1 4 2 9 7
##  [4177] 2 5 1 1 2 1 9 9 9 1 0 2 0 2 1 1 4 6 4 1 5 4 9 7 7 1 5 6 2 2 2 8 0 6 9 6
##  [4213] 1 9 7 7 1 4 8 5 3 4 3 4 9 7 5 0 7 4 8 8 1 5 3 9 5 9 7 6 9 0 3 6 3 9 8 2
##  [4249] 2 1 2 8 6 8 5 5 3 9 4 9 2 5 1 5 1 4 4 1 4 4 3 5 9 1 2 2 3 3 0 2 9 0 0 9
##  [4285] 9 6 0 9 3 2 8 4 1 9 9 7 2 7 9 9 5 9 5 1 1 8 3 5 1 9 5 3 5 4 9 5 9 3 1 9
##  [4321] 0 9 7 5 4 9 2 0 1 0 5 1 4 9 3 3 6 1 5 2 5 2 2 0 9 2 6 6 0 1 2 0 3 0 2 5
##  [4357] 5 7 9 5 5 0 8 9 5 0 3 2 5 9 0 8 8 4 5 8 8 4 5 4 8 5 4 9 2 2 1 2 6 8 8 7
##  [4393] 0 3 6 6 4 3 8 8 7 2 2 0 0 9 3 9 9 1 9 8 6 6 4 2 6 9 2 8 5 4 5 7 9 9 9 2
##  [4429] 1 8 3 4 0 7 8 3 9 3 4 6 5 6 2 3 9 2 6 0 0 6 1 2 8 7 9 8 2 0 4 7 7 5 0 5
##  [4465] 6 4 6 7 4 3 0 7 5 0 7 4 2 0 8 9 9 4 2 4 6 7 8 7 6 9 4 1 3 7 3 0 8 8 7 6
##  [4501] 9 3 9 2 2 9 2 1 8 3 2 9 6 8 4 0 1 2 8 4 5 2 7 8 1 1 3 0 3 5 7 0 3 1 9 3
##  [4537] 6 3 1 7 7 3 0 8 4 8 2 6 5 2 9 7 3 9 0 9 9 6 4 2 9 7 2 1 1 6 7 4 7 5 9 6
##  [4573] 8 2 1 4 4 5 7 6 1 3 2 5 9 9 3 6 1 1 4 6 9 7 2 1 5 1 4 6 3 8 1 1 0 3 1 6
##  [4609] 8 4 9 0 7 3 0 2 9 0 6 6 6 3 6 7 7 2 8 6 0 8 3 0 2 9 8 3 2 5 3 8 8 0 0 1
##  [4645] 9 5 1 3 9 6 0 1 4 1 7 1 2 3 7 9 7 4 9 9 3 9 2 8 2 7 1 8 0 9 1 0 1 7 7 9
##  [4681] 6 9 9 9 2 1 6 1 3 5 7 1 9 7 6 4 5 7 6 6 9 9 6 3 6 2 9 8 1 2 2 5 5 2 3 7
##  [4717] 2 1 0 1 0 4 5 2 8 2 8 3 5 1 7 8 1 1 2 9 7 8 4 0 3 0 7 8 8 4 7 7 8 5 8 4
##  [4753] 9 8 1 3 8 0 3 1 7 9 5 5 1 6 5 7 4 9 3 5 4 7 1 2 0 8 1 6 0 7 3 4 7 3 9 6
##  [4789] 0 8 6 4 8 7 7 9 3 8 6 9 7 2 3 4 0 2 1 8 3 5 5 7 2 4 6 7 2 8 3 0 8 7 8 9
##  [4825] 0 8 4 4 5 8 5 6 6 3 0 9 3 7 6 8 9 3 4 9 5 8 9 1 2 8 8 6 8 1 3 7 9 0 1 1
##  [4861] 4 7 0 8 1 7 4 5 7 1 2 1 1 3 9 6 2 1 2 8 0 7 6 6 9 3 7 0 5 2 8 0 5 4 3 8
##  [4897] 4 6 6 2 7 9 5 1 3 2 4 3 6 1 9 4 4 7 6 5 4 1 9 9 2 7 8 0 1 3 6 1 3 4 1 1
##  [4933] 1 5 6 0 7 0 7 2 3 2 5 2 2 9 4 9 8 1 2 1 6 1 2 7 8 0 0 0 8 2 2 9 2 2 7 9
##  [4969] 9 2 7 5 1 3 4 9 4 1 8 5 6 2 8 3 1 2 8 4 9 9 3 7 0 7 7 2 3 2 4 0 3 9 9 8
##  [5005] 4 1 0 6 0 9 6 8 6 1 1 9 8 9 2 3 5 5 9 4 2 1 9 4 3 9 6 0 4 0 6 0 1 2 3 4
##  [5041] 7 8 9 0 1 2 3 4 7 8 9 0 1 2 3 4 5 6 7 8 9 8 3 4 7 8 6 3 4 0 9 7 1 9 3 8
##  [5077] 4 7 3 0 9 1 4 5 4 6 2 0 6 2 1 1 1 1 7 2 4 7 5 2 9 4 5 8 4 2 9 7 0 0 7 5
##  [5113] 1 1 7 6 6 6 8 2 2 7 7 4 0 2 4 2 1 8 9 6 1 0 5 9 6 9 8 0 3 0 8 3 9 6 3 0
##  [5149] 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 5 4 8 7 4 7 7 3 9 8
##  [5185] 8 3 1 5 8 2 7 4 2 1 5 4 5 5 8 6 4 4 4 1 8 7 5 5 1 8 9 1 3 6 3 3 2 2 6 9
##  [5221] 9 6 5 5 3 3 8 1 6 5 6 8 1 9 7 6 8 3 7 4 7 0 9 0 0 3 7 9 3 0 2 0 1 0 1 0
##  [5257] 4 0 1 0 4 7 9 6 2 6 2 2 9 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
##  [5293] 2 3 4 5 6 7 8 9 8 0 5 6 6 0 8 0 2 3 7 9 4 7 1 9 1 7 1 4 0 0 4 1 7 5 7 1
##  [5329] 3 3 3 1 6 9 7 4 3 0 2 5 2 6 0 8 9 4 3 5 4 8 1 5 9 0 6 4 3 6 3 3 8 1 4 7
##  [5365] 5 7 2 2 0 0 1 7 7 9 5 9 8 9 6 8 8 2 3 6 1 2 9 8 9 5 2 6 2 4 8 4 6 5 0 1
##  [5401] 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7 4 2 0 9 0 1 5 8 8 0
##  [5437] 2 7 8 4 4 6 1 0 4 5 3 9 4 2 0 5 0 1 3 2 9 1 6 0 1 1 8 0 4 7 7 6 3 6 0 7
##  [5473] 3 5 4 2 4 1 8 3 5 6 7 0 6 7 1 2 5 8 1 9 3 8 2 8 7 6 7 1 4 6 2 9 3 0 1 2
##  [5509] 3 4 5 6 7 0 1 2 3 4 5 0 1 2 8 9 1 4 0 9 5 0 8 0 7 7 1 1 2 9 3 6 7 2 3 8
##  [5545] 1 2 9 8 8 7 1 7 1 1 0 3 4 2 6 4 7 4 2 7 4 9 1 0 6 8 5 5 5 3 5 9 7 4 8 5
##  [5581] 9 6 9 3 0 3 8 9 1 8 1 6 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2
##  [5617] 3 4 5 6 7 8 9 3 5 3 2 9 3 2 1 4 5 5 2 3 2 1 3 9 7 2 1 2 8 9 1 8 8 7 8 1
##  [5653] 0 0 7 7 8 7 5 0 6 1 5 7 4 6 1 2 5 0 7 9 9 0 3 8 4 4 8 1 8 6 5 9 0 0 0 3
##  [5689] 7 1 6 4 2 6 6 0 4 5 4 1 3 8 6 3 9 9 5 9 3 7 8 5 6 4 7 6 2 2 0 9 4 0 1 2
##  [5725] 3 4 5 6 7 8 9 0 1 2 3 5 6 0 1 2 3 4 5 6 8 7 1 3 2 8 0 7 5 9 9 6 0 9 4 1
##  [5761] 3 2 1 2 3 8 3 2 6 5 6 8 2 7 4 8 1 8 0 5 3 9 4 1 9 2 1 9 6 7 9 0 4 6 1 7
##  [5797] 3 8 7 2 9 6 5 8 3 9 0 5 7 1 6 1 0 9 3 3 4 4 0 6 2 5 4 2 3 4 6 0 0 2 0 1
##  [5833] 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 8 7 1 3 7 5 2 8 0 7 5
##  [5869] 9 9 0 9 1 1 5 8 8 6 3 2 1 8 3 2 6 5 6 7 4 1 0 5 3 1 9 2 1 9 6 0 4 6 1 7
##  [5905] 3 8 7 2 9 6 5 8 3 5 7 1 6 1 0 9 6 2 5 4 2 3 4 4 6 0 0 2 0 1 2 3 4 5 6 7
##  [5941] 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 8 6 5 0 6 8 9 4 1 9 5 3 0 4
##  [5977] 8 9 1 4 0 5 5 2 1 5 4 0 7 6 0 1 7 0 6 8 9 5 1 7 9 8 6 0 8 1 7 7 1 3 2 3
##  [6013] 1 4 2 0 0 7 8 4 6 4 9 3 8 4 7 2 5 6 3 6 9 6 3 2 2 4 6 9 0 2 5 5 1 3 3 9
##  [6049] 7 8 7 2 2 5 7 9 8 2 1 3 1 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
##  [6085] 2 3 4 5 6 7 8 9 1 2 6 5 3 0 7 0 4 1 4 3 6 7 2 3 1 2 1 2 9 6 0 1 3 0 2 7
##  [6121] 5 7 6 2 9 1 9 0 6 0 6 0 2 0 6 1 5 8 4 3 0 1 5 4 4 8 5 7 5 7 8 3 4 8 8 5
##  [6157] 2 9 7 1 3 8 1 0 7 5 9 6 9 4 7 7 9 9 3 4 4 3 8 6 2 0 1 2 3 4 5 6 7 8 9 0
##  [6193] 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 8 3 9 5 5 2 6 8 4 9 1 7 1 2 3 5
##  [6229] 9 6 9 1 1 1 2 9 5 6 8 1 2 0 7 7 5 8 2 9 8 9 0 4 6 7 1 3 4 5 6 0 3 6 8 7
##  [6265] 0 4 2 7 4 7 5 4 3 4 2 8 1 5 1 2 0 2 5 6 4 3 0 0 0 3 3 5 7 0 6 4 8 8 6 3
##  [6301] 4 6 9 9 8 2 7 7 1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7
##  [6337] 8 2 1 7 2 5 0 8 0 2 7 8 8 3 6 0 2 7 6 6 1 2 8 8 7 7 4 7 7 3 7 4 5 4 3 3
##  [6373] 8 4 1 1 9 7 4 3 7 3 3 0 2 5 5 6 6 3 5 2 5 9 9 8 4 1 0 6 0 9 6 8 8 5 6 1
##  [6409] 1 9 8 9 2 3 5 5 9 4 2 1 9 3 9 2 0 6 0 4 0 0 1 2 3 4 7 8 9 0 1 2 3 7 8 9
##  [6445] 0 1 2 3 4 7 8 9 7 3 0 3 1 8 7 6 4 0 2 6 8 3 2 8 1 2 0 7 1 0 4 4 5 8 0 6
##  [6481] 2 3 1 5 1 8 5 9 4 0 7 5 8 8 3 8 9 2 6 2 5 3 1 7 3 9 1 9 9 6 0 3 9 2 8 1
##  [6517] 4 3 5 2 9 2 5 8 9 5 0 1 2 4 5 6 0 1 2 3 4 5 6 7 1 2 3 4 5 1 0 4 5 6 6 3
##  [6553] 4 4 2 8 1 0 6 4 9 7 2 3 3 9 2 0 9 3 3 9 1 5 2 3 7 7 8 4 0 2 4 0 2 4 7 8
##  [6589] 0 7 0 6 9 3 2 8 6 0 5 7 5 1 0 8 1 6 7 2 9 7 9 5 8 6 2 6 2 8 1 7 5 0 1 1
##  [6625] 3 8 4 9 1 8 6 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 7 8 9 0 1 7 8 9 9 8 9 8
##  [6661] 4 1 7 7 3 3 7 6 6 6 1 9 0 1 7 6 3 2 1 7 1 3 9 1 7 6 8 4 1 4 3 6 9 6 1 4
##  [6697] 4 7 2 4 4 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 9 0 1 2 3 4 7 8 1 3 5 1 7 7
##  [6733] 2 1 4 8 3 4 4 3 9 7 4 1 2 3 5 9 1 6 0 1 0 0 2 8 7 1 1 4 0 4 7 3 6 8 0 3
##  [6769] 7 4 0 6 9 2 6 5 8 6 9 0 4 0 6 1 9 2 0 9 5 1 3 7 6 9 3 0 2 2 0 1 2 3 4 5
##  [6805] 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 2 1 7 2 5 0 8 0 2 7 8 8
##  [6841] 3 0 6 0 2 7 6 6 1 2 8 8 7 7 4 7 7 3 7 4 5 4 3 3 8 4 5 4 1 1 9 7 4 3 7 3
##  [6877] 3 0 2 5 5 6 3 1 5 2 5 9 9 8 4 1 0 6 0 9 6 8 8 5 6 1 1 9 8 9 2 3 5 5 9 4
##  [6913] 2 1 9 4 9 1 3 9 2 0 6 0 4 0 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
##  [6949] 1 2 3 4 5 6 7 8 9 3 8 0 7 1 0 7 5 5 6 9 0 1 0 0 8 3 4 3 1 5 0 0 9 5 3 4
##  [6985] 9 3 7 6 9 2 4 5 7 2 6 4 9 4 9 4 1 2 2 5 8 1 3 2 9 4 3 8 2 2 1 2 8 6 5 1
##  [7021] 6 7 2 1 3 9 3 8 7 5 7 0 7 4 8 8 5 0 6 6 3 7 6 9 9 4 8 4 1 0 6 6 0 1 2 3
##  [7057] 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7 4 0 4 0 1 7 9 5 1
##  [7093] 4 2 8 9 4 3 7 8 2 4 4 3 3 6 9 9 5 8 6 7 0 6 8 2 6 3 9 3 2 8 6 1 7 4 8 8
##  [7129] 9 0 3 3 9 0 5 2 9 4 1 0 3 7 5 8 7 7 8 2 9 7 1 2 6 4 2 5 2 3 6 6 5 0 0 2
##  [7165] 8 1 6 1 0 4 3 1 6 1 9 0 1 4 5 6 7 8 9 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9
##  [7201] 8 4 0 0 7 2 4 3 8 6 6 3 2 6 3 3 0 1 4 7 8 0 3 1 9 0 1 9 1 2 7 0 1 3 8 2
##  [7237] 9 2 7 6 5 5 9 9 8 2 9 1 3 2 3 4 3 1 9 0 9 3 6 8 7 0 1 0 5 8 2 7 7 0 1 2
##  [7273] 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 7 4 8 1 5 6 5 7
##  [7309] 2 8 6 3 3 8 6 5 4 0 9 1 7 2 9 1 5 1 3 2 2 3 0 6 4 3 7 6 9 0 4 8 1 4 0 6
##  [7345] 1 2 6 9 2 2 3 5 5 1 0 7 7 9 6 2 9 4 7 0 2 3 4 0 0 8 8 8 5 1 3 7 4 9 8 8
##  [7381] 9 0 9 8 9 0 2 6 5 6 7 4 7 5 4 1 3 5 3 1 2 3 4 5 6 1 2 3 4 6 0 1 2 4 5 6
##  [7417] 7 8 1 7 2 4 1 4 1 4 9 6 8 4 5 3 7 8 4 3 3 5 6 7 0 6 1 6 8 7 0 1 5 0 8 5
##  [7453] 0 1 5 8 4 2 3 9 7 6 9 1 9 0 6 7 1 2 3 9 2 4 5 5 3 7 5 3 1 8 2 2 3 0 2 9
##  [7489] 4 9 7 0 2 7 4 9 9 2 5 9 8 3 8 6 7 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7
##  [7525] 8 9 0 1 2 3 4 5 6 7 8 9 0 0 7 2 6 5 5 3 7 8 6 6 6 6 4 3 8 8 3 0 1 9 0 5
##  [7561] 4 1 9 1 2 7 0 1 3 8 2 9 2 7 4 2 6 5 5 9 9 1 1 5 7 6 8 2 9 4 3 1 9 0 9 3
##  [7597] 6 8 7 0 1 0 5 8 2 7 7 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 8 9 0 1 2 3 4 5 6
##  [7633] 7 8 9 2 1 2 1 3 9 9 8 5 3 7 0 7 7 5 7 9 9 4 7 0 3 4 1 5 8 1 4 8 4 1 8 6
##  [7669] 6 4 6 0 5 5 3 3 5 7 2 5 9 6 9 2 6 2 1 2 0 8 3 8 3 0 8 7 4 9 5 0 9 7 0 0
##  [7705] 4 6 0 9 1 6 2 7 6 8 3 5 2 1 8 3 8 6 1 0 2 1 4 0 1 2 3 4 5 6 7 8 9 0 1 2
##  [7741] 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7 6 4 7 6 2 3 4 8 7 8 6 9 8 3 2 2 8 4
##  [7777] 8 5 6 5 0 2 0 1 1 2 9 6 8 2 1 0 6 5 2 9 7 5 3 9 3 7 1 8 3 8 1 9 5 5 0 1
##  [7813] 1 9 8 2 6 0 4 5 0 3 1 8 6 7 5 9 9 3 0 3 1 4 4 0 4 9 0 1 2 3 5 6 7 8 0 1
##  [7849] 2 3 5 6 7 8 9 0 1 2 3 5 6 7 8 9 9 7 0 9 0 1 5 8 8 0 9 3 2 7 8 4 6 1 0 4
##  [7885] 9 4 2 0 5 0 1 6 9 3 2 9 1 6 0 1 1 8 7 7 6 3 6 0 7 2 4 1 7 0 6 7 1 2 5 8
##  [7921] 1 8 2 8 7 6 8 7 1 6 2 9 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2
##  [7957] 3 4 5 6 7 8 9 8 9 5 7 0 3 1 6 8 4 1 5 6 4 2 7 8 1 3 4 3 4 7 2 0 5 0 1 9
##  [7993] 2 3 2 3 5 5 7 8 4 9 9 7 1 1 9 0 7 8 3 4 8 6 3 8 0 9 6 2 1 0 1 0 6 2 3 8
##  [8029] 9 0 7 2 3 4 5 5 2 8 5 4 6 6 6 7 9 1 8 2 1 5 3 4 7 9 4 0 0 0 1 2 3 4 5 6
##  [8065] 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 9 0 1 3 1 5 1 2 4 9 2 4 6 8 0 1
##  [8101] 1 9 2 6 6 8 7 4 2 9 7 0 2 1 0 3 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8
##  [8137] 9 0 1 2 3 4 5 6 7 8 9 8 6 5 9 7 0 2 3 4 3 8 5 1 5 2 3 0 1 2 1 3 2 6 5 3
##  [8173] 0 7 2 7 4 6 4 0 5 9 9 8 9 5 3 1 7 4 7 6 5 4 0 0 6 6 2 0 6 3 7 7 4 4 3 9
##  [8209] 2 8 9 6 0 9 5 3 8 8 7 1 4 0 4 8 5 2 3 9 0 1 9 1 5 1 7 4 8 6 2 1 6 8 8 0
##  [8245] 1 2 3 4 7 8 9 0 1 2 3 4 6 7 8 9 0 1 2 3 4 7 8 9 1 4 5 3 3 0 9 5 4 3 0 8
##  [8281] 4 6 7 0 7 7 1 6 9 1 3 6 2 3 8 2 3 8 9 5 8 8 7 1 7 1 1 0 3 4 2 6 4 7 4 2
##  [8317] 7 4 2 9 2 7 9 2 1 0 6 5 3 4 8 5 9 6 9 0 6 3 0 8 1 6 0 0 1 2 3 4 5 6 7 0
##  [8353] 1 2 3 4 7 8 9 0 1 2 3 4 7 2 5 1 6 4 3 9 9 0 9 7 1 6 4 3 6 2 0 9 8 6 5 7
##  [8389] 0 0 1 7 4 3 2 4 1 3 7 6 4 7 7 7 9 8 4 3 8 2 8 3 5 8 0 5 4 7 1 3 1 7 9 6
##  [8425] 2 0 9 1 7 3 3 9 1 6 4 3 9 8 2 1 8 6 4 1 5 5 6 5 0 1 2 3 4 5 6 7 8 9 0 1
##  [8461] 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 6 9 7 0 2 3 4 3 8 5 1 3 0 1 2 1 3 2
##  [8497] 0 7 2 6 4 0 5 9 9 8 9 5 3 1 7 4 7 0 0 6 6 6 3 7 4 2 8 9 8 7 1 4 0 4 8 5
##  [8533] 2 3 9 0 1 9 1 5 1 7 6 1 2 1 6 8 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 0
##  [8569] 1 2 3 5 6 7 8 1 0 4 5 6 6 3 4 4 2 8 1 0 6 4 9 7 2 9 2 0 9 3 3 9 1 5 2 3
##  [8605] 1 6 7 3 7 8 4 0 2 4 0 2 4 7 8 0 7 0 6 9 3 2 4 8 6 0 5 7 5 1 0 8 1 6 7 2
##  [8641] 9 7 9 5 6 5 2 6 2 8 1 7 5 5 7 3 5 0 1 1 3 8 4 9 4 5 1 8 6 8 9 0 1 2 3 4
##  [8677] 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 3 5 3 2 9 3 2 1 4 5 5
##  [8713] 2 3 2 1 3 9 7 2 1 2 8 9 1 8 8 7 8 1 0 0 6 7 7 8 7 5 0 6 1 5 7 4 6 1 2 5
##  [8749] 0 7 9 9 0 3 4 4 8 4 1 8 6 5 9 0 0 0 3 7 1 6 4 6 0 4 5 4 1 3 8 6 3 9 9 5
##  [8785] 9 3 7 8 5 6 4 7 6 2 2 0 9 4 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
##  [8821] 2 3 4 5 6 7 8 9 6 4 2 6 4 7 5 5 4 7 2 9 3 9 3 8 2 0 9 5 6 0 1 0 6 5 3 5
##  [8857] 3 8 0 0 3 4 1 5 3 0 8 3 0 6 2 7 8 1 7 1 3 8 5 4 2 0 9 7 6 7 4 1 6 2 6 7
##  [8893] 1 9 8 0 6 9 4 9 9 6 2 3 7 1 9 2 2 5 3 7 8 0 1 2 3 4 7 8 9 0 1 2 3 4 7 8
##  [8929] 9 0 1 7 8 9 8 9 2 6 1 3 5 4 8 2 6 4 3 4 5 9 2 0 3 9 4 9 7 3 8 7 4 4 9 8
##  [8965] 5 8 2 6 6 2 3 1 3 2 7 3 1 9 0 1 1 3 5 0 7 8 1 5 1 4 6 0 0 4 9 1 6 6 9 0
##  [9001] 7 6 1 1 0 1 2 3 4 7 2 3 4 5 6 7 0 1 2 7 8 6 3 9 7 1 9 3 9 6 1 7 2 4 4 5
##  [9037] 7 0 0 1 6 6 8 2 7 7 2 4 2 1 6 1 0 6 9 8 3 9 6 3 0 1 2 3 4 5 6 7 8 9 0 1
##  [9073] 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 6 8 9 9 0 1 2 4 4 3 7 4 4 4 0 3 8
##  [9109] 7 5 8 2 1 7 5 3 8 5 2 5 1 1 6 2 1 3 8 6 4 2 6 2 5 5 0 2 8 0 6 8 1 7 9 1
##  [9145] 9 2 6 7 6 6 8 7 4 9 2 1 3 3 0 5 5 8 0 3 7 9 7 0 2 7 9 1 7 8 0 3 5 3 6 0
##  [9181] 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 7 8 9 6 4 2 6 4 7 8 9 2
##  [9217] 9 3 9 3 0 0 1 0 4 2 6 3 5 3 0 3 4 1 5 3 0 8 3 0 6 1 7 8 0 9 2 6 7 1 9 6
##  [9253] 9 4 9 9 6 7 1 2 5 3 7 8 0 1 2 4 5 6 7 8 9 0 1 3 4 5 6 7 8 0 1 3 4 7 8 9
##  [9289] 7 5 5 1 9 9 7 1 0 0 5 9 7 1 7 2 2 3 6 8 3 2 0 0 6 1 7 5 8 6 2 9 4 8 8 7
##  [9325] 1 0 8 7 7 5 8 5 3 4 6 1 1 5 5 0 7 2 3 6 4 1 2 4 1 5 4 2 0 4 8 6 1 9 0 2
##  [9361] 5 6 9 3 6 3 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 5 6 7 8 1
##  [9397] 0 9 5 7 5 1 8 6 9 0 4 1 9 3 8 4 4 7 0 1 9 2 8 7 8 2 5 9 6 0 6 5 5 3 3 3
##  [9433] 9 8 1 1 0 6 1 0 0 6 2 1 1 3 2 7 7 8 8 7 8 4 6 0 2 0 7 0 3 6 8 7 1 5 9 9
##  [9469] 3 7 2 4 9 4 3 6 2 2 5 3 2 5 5 9 4 1 7 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
##  [9505] 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 0 1 2 7 5 3 4 4 0 0 6 9 6 6 5 7 2 3 4 4 9
##  [9541] 1 4 0 7 9 5 7 2 3 1 4 4 0 9 9 6 1 8 3 3 7 3 9 8 8 4 7 7 6 2 1 9 8 7 8 8
##  [9577] 7 2 2 3 9 3 3 5 5 0 7 9 5 6 5 1 4 1 1 2 8 2 6 1 5 0 1 2 3 4 5 6 7 8 9 0
##  [9613] 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 8 0 6 0 0 2 3 7 9 4 7 1 9 1 7 1 4 0
##  [9649] 0 1 7 5 7 1 3 3 3 1 6 9 7 1 3 0 2 6 0 8 9 4 3 5 4 8 1 5 9 0 6 6 3 8 1 4
##  [9685] 7 5 2 0 0 1 7 8 9 6 8 8 2 3 6 1 2 9 5 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
##  [9721] 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7 4 6 1 4 0 9 9 3 7 8 4 7 5 8 5 3 2 2 0 5 8
##  [9757] 6 0 3 8 1 0 3 0 4 7 4 9 2 9 5 7 1 7 1 6 6 5 6 2 8 7 6 4 9 9 5 3 7 4 3 0
##  [9793] 4 6 6 1 1 3 2 1 0 0 1 2 3 4 7 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 7 8 9 0 8
##  [9829] 3 9 5 5 2 6 8 4 1 7 1 2 3 5 6 9 1 1 1 2 1 2 0 7 7 5 8 2 9 8 6 7 3 4 6 8
##  [9865] 7 0 4 2 7 7 5 4 3 4 2 8 1 5 1 0 2 3 3 5 7 0 6 8 6 3 9 9 8 2 7 7 1 0 1 7
##  [9901] 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 7 8 9 7 8 6 4 1 9 3 8 4 4 7 0 1 9 2 8 7
##  [9937] 8 2 6 0 6 5 3 3 3 9 1 4 0 6 1 0 0 6 2 1 1 7 7 8 4 6 0 7 0 3 6 8 7 1 5 2
##  [9973] 4 9 4 3 6 4 1 7 2 6 5 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6

cor(prognoze,g_test)

## [1] 0.9773665

In logistics, accurate digit classification (e.g., postal codes) is crucial for optimizing delivery processes and ensuring correct package routing.

For simpler tasks, we can use a multiclass logistic regression model, which is faster but less complex than the previous neural network.

modellr <- keras_model_sequential() %>%
  layer_dense(input_shape = 784, units = 10,
       activation = "softmax")
summary(modellr)

## Model: "sequential_1"
## ________________________________________________________________________________
##  Layer (type)                       Output Shape                    Param #     
## ================================================================================
##  dense_3 (Dense)                    (None, 10)                      7850        
## ================================================================================
## Total params: 7850 (30.66 KB)
## Trainable params: 7850 (30.66 KB)
## Non-trainable params: 0 (0.00 Byte)
## ________________________________________________________________________________

The training process is similar, but the architecture is simpler, making it faster for less demanding tasks.

modellr %>% compile(loss = "categorical_crossentropy",
     optimizer = optimizer_rmsprop(), metrics = c("accuracy"))
modellr %>% fit(x_train, y_train, epochs = 30,
      batch_size = 128, validation_split = 0.2)

## Epoch 1/30
## 375/375 - 1s - loss: 0.6791 - accuracy: 0.8339 - val_loss: 0.3593 - val_accuracy: 0.9055 - 1s/epoch - 4ms/step
## Epoch 2/30
## 375/375 - 1s - loss: 0.3530 - accuracy: 0.9039 - val_loss: 0.3098 - val_accuracy: 0.9153 - 903ms/epoch - 2ms/step
## Epoch 3/30
## 375/375 - 1s - loss: 0.3177 - accuracy: 0.9116 - val_loss: 0.2923 - val_accuracy: 0.9197 - 907ms/epoch - 2ms/step
## Epoch 4/30
## 375/375 - 1s - loss: 0.3020 - accuracy: 0.9158 - val_loss: 0.2864 - val_accuracy: 0.9221 - 901ms/epoch - 2ms/step
## Epoch 5/30
## 375/375 - 1s - loss: 0.2923 - accuracy: 0.9184 - val_loss: 0.2784 - val_accuracy: 0.9235 - 893ms/epoch - 2ms/step
## Epoch 6/30
## 375/375 - 1s - loss: 0.2857 - accuracy: 0.9204 - val_loss: 0.2743 - val_accuracy: 0.9246 - 900ms/epoch - 2ms/step
## Epoch 7/30
## 375/375 - 1s - loss: 0.2807 - accuracy: 0.9225 - val_loss: 0.2718 - val_accuracy: 0.9259 - 929ms/epoch - 2ms/step
## Epoch 8/30
## 375/375 - 1s - loss: 0.2766 - accuracy: 0.9236 - val_loss: 0.2731 - val_accuracy: 0.9262 - 952ms/epoch - 3ms/step
## Epoch 9/30
## 375/375 - 1s - loss: 0.2732 - accuracy: 0.9240 - val_loss: 0.2680 - val_accuracy: 0.9276 - 949ms/epoch - 3ms/step
## Epoch 10/30
## 375/375 - 1s - loss: 0.2707 - accuracy: 0.9248 - val_loss: 0.2671 - val_accuracy: 0.9272 - 965ms/epoch - 3ms/step
## Epoch 11/30
## 375/375 - 1s - loss: 0.2685 - accuracy: 0.9263 - val_loss: 0.2661 - val_accuracy: 0.9268 - 941ms/epoch - 3ms/step
## Epoch 12/30
## 375/375 - 1s - loss: 0.2667 - accuracy: 0.9266 - val_loss: 0.2665 - val_accuracy: 0.9279 - 949ms/epoch - 3ms/step
## Epoch 13/30
## 375/375 - 1s - loss: 0.2649 - accuracy: 0.9279 - val_loss: 0.2642 - val_accuracy: 0.9288 - 946ms/epoch - 3ms/step
## Epoch 14/30
## 375/375 - 1s - loss: 0.2634 - accuracy: 0.9277 - val_loss: 0.2633 - val_accuracy: 0.9292 - 987ms/epoch - 3ms/step
## Epoch 15/30
## 375/375 - 1s - loss: 0.2618 - accuracy: 0.9278 - val_loss: 0.2646 - val_accuracy: 0.9288 - 943ms/epoch - 3ms/step
## Epoch 16/30
## 375/375 - 1s - loss: 0.2605 - accuracy: 0.9283 - val_loss: 0.2648 - val_accuracy: 0.9294 - 943ms/epoch - 3ms/step
## Epoch 17/30
## 375/375 - 1s - loss: 0.2596 - accuracy: 0.9290 - val_loss: 0.2627 - val_accuracy: 0.9313 - 944ms/epoch - 3ms/step
## Epoch 18/30
## 375/375 - 1s - loss: 0.2584 - accuracy: 0.9287 - val_loss: 0.2618 - val_accuracy: 0.9311 - 931ms/epoch - 2ms/step
## Epoch 19/30
## 375/375 - 1s - loss: 0.2573 - accuracy: 0.9294 - val_loss: 0.2644 - val_accuracy: 0.9292 - 902ms/epoch - 2ms/step
## Epoch 20/30
## 375/375 - 1s - loss: 0.2564 - accuracy: 0.9302 - val_loss: 0.2626 - val_accuracy: 0.9295 - 898ms/epoch - 2ms/step
## Epoch 21/30
## 375/375 - 1s - loss: 0.2554 - accuracy: 0.9302 - val_loss: 0.2633 - val_accuracy: 0.9298 - 898ms/epoch - 2ms/step
## Epoch 22/30
## 375/375 - 1s - loss: 0.2547 - accuracy: 0.9300 - val_loss: 0.2631 - val_accuracy: 0.9295 - 1s/epoch - 3ms/step
## Epoch 23/30
## 375/375 - 1s - loss: 0.2540 - accuracy: 0.9305 - val_loss: 0.2615 - val_accuracy: 0.9308 - 1s/epoch - 3ms/step
## Epoch 24/30
## 375/375 - 1s - loss: 0.2532 - accuracy: 0.9314 - val_loss: 0.2640 - val_accuracy: 0.9307 - 940ms/epoch - 3ms/step
## Epoch 25/30
## 375/375 - 1s - loss: 0.2525 - accuracy: 0.9309 - val_loss: 0.2626 - val_accuracy: 0.9308 - 914ms/epoch - 2ms/step
## Epoch 26/30
## 375/375 - 1s - loss: 0.2520 - accuracy: 0.9315 - val_loss: 0.2640 - val_accuracy: 0.9307 - 939ms/epoch - 3ms/step
## Epoch 27/30
## 375/375 - 1s - loss: 0.2513 - accuracy: 0.9316 - val_loss: 0.2636 - val_accuracy: 0.9293 - 959ms/epoch - 3ms/step
## Epoch 28/30
## 375/375 - 1s - loss: 0.2506 - accuracy: 0.9318 - val_loss: 0.2627 - val_accuracy: 0.9304 - 940ms/epoch - 3ms/step
## Epoch 29/30
## 375/375 - 1s - loss: 0.2497 - accuracy: 0.9319 - val_loss: 0.2626 - val_accuracy: 0.9314 - 999ms/epoch - 3ms/step
## Epoch 30/30
## 375/375 - 1s - loss: 0.2494 - accuracy: 0.9316 - val_loss: 0.2624 - val_accuracy: 0.9303 - 909ms/epoch - 2ms/step

modellr %>% predict(x_test) %>% k_argmax() %>% accuracy(g_test)

## 313/313 - 0s - 426ms/epoch - 1ms/step

## [1] 0.9284

While this approach is faster, it may not be as effective on complex datasets like MNIST. However, it can still be useful for logistics applications where digit recognition is less complex but still necessary.