CF1955B Progressive Square

A progressive square of size $ n $ is an $ n \times n $ matrix. Maxim chooses three integers $ a_{1,1} $ , $ c $ , and $ d $ and constructs a progressive square according to the following rules: $ $$$a_{i+1,j} = a_{i,j} + c $ $

$ $ a_{i,j+1} = a_{i,j} + d $ $

For example, if $ n = 3 $ , $ a\_{1,1} = 1 $ , $ c=2 $ , and $ d=3 $ , then the progressive square looks as follows:

$ $ \begin{pmatrix} 1 & 4 & 7 \\ 3 & 6 & 9 \\ 5 & 8 & 11 \end{pmatrix} $ $

Last month Maxim constructed a progressive square and remembered the values of $ n $ , $ c $ , and $ d $ . Recently, he found an array $ b $ of $ n^2 $ integers in random order and wants to make sure that these elements are the elements of that specific square.

It can be shown that for any values of $ n $ , $ a\_{1,1} $ , $ c $ , and $ d$$$, there exists exactly one progressive square that satisfies all the rules.

The first line contains an integer $ t $ ( $ 1 \le t \le {10} ^ 4 $ ) — the number of test cases. The first line of each test case contains three integers $ n $ , $ c $ , and $ d $ ( $ 2 \le n \le 500 $ , $ 1 \le c, d \le 10^6 $ ) — the size of the square and the values of $ c $ and $ d $ as described in the statement. The second line of each test case contains $ n \cdot n $ integers $ b_1, b_2, \dots, b_{n \cdot n} $ ( $ 1 \le b_i \le 10^9 $ ) — the elements found by Maxim. It is guaranteed that the sum of $ n ^ 2 $ over all test cases does not exceed $ 25 \cdot {10} ^ 4 $ .

For each test case, output "YES" in a separate line if a progressive square for the given $ n $ , $ c $ , and $ d $ can be constructed from the array elements $ a $ , otherwise output "NO". You can output each letter in any case (lowercase or uppercase). For example, the strings "yEs", "yes", "Yes", and "YES" will be accepted as a positive answer.