Score : $500$ points

Problem Statement

Given are an $N \times N$ matrix and an integer $K$. The entry in the $i$-th row and $j$-th column of this matrix is denoted as $a_{i, j}$. This matrix contains each of $1, 2, \dots, N^2$ exactly once.

Sigma can repeat the following two kinds of operation arbitrarily many times in any order.

• Pick two integers $x, y (1 \leq x < y \leq N)$ that satisfy $a_{i, x} + a_{i, y} \leq K$ for all $i$ ($1 \leq i \leq N$) and swap the $x$-th and the $y$-th columns.
• Pick two integers $x, y (1 \leq x < y \leq N)$ that satisfy $a_{x, i} + a_{y, i} \leq K$ for all $i$ ($1 \leq i \leq N$) and swap the $x$-th and the $y$-th rows.

How many matrices can he obtain by these operations? Find it modulo $998244353$.

Constraints

• $1 \leq N \leq 50$
• $1 \leq K \leq 2 \times N^2$
• $a_{i, j}$'s are a rearrangement of $1, 2, \dots, N^2$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$

$a_{1, 1}$ $a_{1, 2}$ $...$ $a_{1, N}$

$a_{2, 1}$ $a_{2, 2}$ $...$ $a_{2, N}$

$:$

$a_{N, 1}$ $a_{N, 2}$ $...$ $a_{N, N}$

Output

Print the number of matrices Sigma can obtain modulo $998244353$.

3 13
3 2 7
4 8 9
1 6 5

12

For example, Sigma can swap two columns, by setting $x = 1, y = 2$. After that, the resulting matrix will be:

2 3 7
8 4 9
6 1 5

After that, he can swap two row vectors by setting $x = 1, y = 3$, resulting in the following matrix:

6 1 5
8 4 9
2 3 7

10 165
82 94 21 65 28 22 61 80 81 79
93 35 59 85 96 1 78 72 43 5
12 15 97 49 69 53 18 73 6 58
60 14 23 19 44 99 64 17 29 67
24 39 56 92 88 7 48 75 36 91
74 16 26 10 40 63 45 76 86 3
9 66 42 84 38 51 25 2 33 41
87 54 57 62 47 31 68 11 83 8
46 27 55 70 52 98 20 77 89 34
32 71 30 50 90 4 37 95 13 100

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