Score : $100$ points

Problem Statement

There is a simple directed graph $G$ with $N$ vertices, numbered $1, 2, \ldots, N$.

For each $i$ and $j$ ($1 \leq i, j \leq N$), you are given an integer $a_{i, j}$ that represents whether there is a directed edge from Vertex $i$ to $j$. If $a_{i, j} = 1$, there is a directed edge from Vertex $i$ to $j$; if $a_{i, j} = 0$, there is not.

Find the number of different directed paths of length $K$ in $G$, modulo $10^9 + 7$. We will also count a path that traverses the same edge multiple times.

Constraints

• All values in input are integers.
• $1 \leq N \leq 50$
• $1 \leq K \leq 10^{18}$
• $a_{i, j}$ is $0$ or $1$.
• $a_{i, i} = 0$

Input

Input is given from Standard Input in the following format:

$N$ $K$

$a_{1, 1}$ $\ldots$ $a_{1, N}$

$:$

$a_{N, 1}$ $\ldots$ $a_{N, N}$

Output

Print the number of different directed paths of length $K$ in $G$, modulo $10^9 + 7$.

4 2
0 1 0 0
0 0 1 1
0 0 0 1
1 0 0 0

6

$G$ is drawn in the figure below:

There are six directed paths of length $2$:

• $1$ → $2$ → $3$
• $1$ → $2$ → $4$
• $2$ → $3$ → $4$
• $2$ → $4$ → $1$
• $3$ → $4$ → $1$
• $4$ → $1$ → $2$

3 3
0 1 0
1 0 1
0 0 0

3

$G$ is drawn in the figure below:

There are three directed paths of length $3$:

• $1$ → $2$ → $1$ → $2$
• $2$ → $1$ → $2$ → $1$
• $2$ → $1$ → $2$ → $3$

6 2
0 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 0 0

1

$G$ is drawn in the figure below:

There is one directed path of length $2$:

• $4$ → $5$ → $6$

1 1
0

0

10 1000000000000000000
0 0 1 1 0 0 0 1 1 0
0 0 0 0 0 1 1 1 0 0
0 1 0 0 0 1 0 1 0 1
1 1 1 0 1 1 0 1 1 0
0 1 1 1 0 1 0 1 1 1
0 0 0 1 0 0 1 0 1 0
0 0 0 1 1 0 0 1 0 1
1 0 0 0 1 0 1 0 0 0
0 0 0 0 0 1 0 0 0 0
1 0 1 1 1 0 1 1 1 0

957538352

Be sure to print the count modulo $10^9 + 7$.