Progressive Square
Description
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.
Input
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 $25 \cdot {10} ^ 4$.
Output
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.
5
3 2 3
3 9 6 5 7 1 10 4 8
3 2 3
3 9 6 5 7 1 11 4 8
2 100 100
400 300 400 500
3 2 3
3 9 6 6 5 1 11 4 8
4 4 4
15 27 7 19 23 23 11 15 7 3 19 23 11 15 11 15
NO
YES
YES
NO
NO