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 ai,j​. This matrix contains each of 1,2,…,N2 exactly once.

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

• Pick two integers $x,y(1\le x<y\le N)$ that satisfy ai,x​+ai,y​≤K for all $i$ ($1\le i\le N$) and swap the $x$-th and the $y$-th columns.
• Pick two integers $x,y(1\le x<y\le N)$ that satisfy ax,i​+ay,i​≤K for all $i$ ($1\le i\le N$) and swap the $x$-th and the $y$-th rows.

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

Constraints

• $1\le N\le 50$
• 1≤K≤2×N2
• ai,j​'s are a rearrangement of 1,2,…,N2.
• All values in input are integers.

Sample Input 1

3 13
3 2 7
4 8 9
1 6 5

Sample Output 1

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

Sample Input 2

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

Sample Output 2

348179577