CF1801A The Very Beautiful Blanket

Kirill wants to weave the very beautiful blanket consisting of $ n \times m $ of the same size square patches of some colors. He matched some non-negative integer to each color. Thus, in our problem, the blanket can be considered a $ B $ matrix of size $ n \times m $ consisting of non-negative integers. Kirill considers that the blanket is very beautiful, if for each submatrix $ A $ of size $ 4 \times 4 $ of the matrix $ B $ is true: - $ A_{11} \oplus A_{12} \oplus A_{21} \oplus A_{22} = A_{33} \oplus A_{34} \oplus A_{43} \oplus A_{44}, $ - $ A_{13} \oplus A_{14} \oplus A_{23} \oplus A_{24} = A_{31} \oplus A_{32} \oplus A_{41} \oplus A_{42}, $ where $ \oplus $ means [bitwise exclusive OR](https://en.wikipedia.org/wiki/Exclusive_or) Kirill asks you to help her weave a very beautiful blanket, and as colorful as possible! He gives you two integers $ n $ and $ m $ . Your task is to generate a matrix $ B $ of size $ n \times m $ , which corresponds to a very beautiful blanket and in which the number of different numbers maximized.

The first line of input data contains one integer number $ t $ ( $ 1 \le t \le 1000 $ ) — the number of test cases. The single line of each test case contains two integers $ n $ and $ m $ $ (4 \le n, \, m \le 200) $ — the size of matrix $ B $ . It is guaranteed that the sum of $ n \cdot m $ does not exceed $ 2 \cdot 10^5 $ .

For each test case, in first line output one integer $ cnt $ $ (1 \le cnt \le n \cdot m) $ — the maximum number of different numbers in the matrix. Then output the matrix $ B $ $ (0 \le B_{ij} < 2^{63}) $ of size $ n \times m $ . If there are several correct matrices, it is allowed to output any one. It can be shown that if there exists a matrix with an optimal number of distinct numbers, then there exists among suitable matrices such a $ B $ that $ (0 \le B_{ij} < 2^{63}) $ .

In the first test case, there is only 4 submatrix of size $ 4 \times 4 $ . Consider a submatrix whose upper-left corner coincides with the upper-left corner of the matrix $ B $ : $ \left[ {\begin{array}{cccc} 9740 & 1549 & 9744 & 1553 \\ 1550 & 1551 & 1554 & 1555 \\ 10252 & 2061 & 10256 & 2065 \\ 2062 & 2063 & 2066 & 2067 \\ \end{array} } \right] $ $ 9740 \oplus 1549 \oplus 1550 \oplus 1551 $ $ = $ $ 10256 \oplus 2065 \oplus 2066 \oplus 2067 $ $ = $ $ 8192 $ ; $ 10252 \oplus 2061 \oplus 2062 \oplus 2063 $ $ = $ $ 9744 \oplus 1553 \oplus 1554 \oplus 1555 $ $ = $ $ 8192 $ . So, chosen submatrix fits the condition. Similarly, you can make sure that the other three submatrices also fit the condition.