Problem:
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.

Sample Input 1

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

Sample Output 1

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

Sample Input 2

3 3
0 1 0
1 0 1
0 0 0

Sample Output 2

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

Sample Input 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

Sample Output 3

1

G is drawn in the figure below:

There is one directed path of length 2:

4 → 5 → 6

Sample Input 4

1 1
0

Sample Output 4

0

Sample Input 5

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

Sample Output 5

957538352

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

Solution in Python (3.4.3) (judge verdict: Time Limit Exceeded)

MOD = 10 ** 9 + 7

N, K = map(int, input().split())

A = [list(map(int, input().split())) for _ in range(N)]

def mod_pow(matrix, n, mod): #行列累乗を求める
    # print (matrix)
    N = len(matrix[0])
    if n == 0:
        lst = [[1 if i == j else 0 for i in range(N)] for j in range(N)]
        return lst
    lst = [[0] * N for _ in range(N)]
    for i in range(N):
        for j in range(N):
            for k in range(N):
                # print (i, j, k)
                lst[i][j] += matrix[i][k] * matrix[k][j]
            lst[i][j] %= mod
    res = mod_pow(lst, n // 2, mod)

    lst = [[0] * N for _ in range(N)]
    if n % 2 == 1:
        for i in range(N):
            for j in range(N):
                for k in range(N):
                    lst[i][j] += res[i][k] * matrix[k][j]
                lst[i][j] %= mod
        return lst
    return res 

matrix = mod_pow(A, K, MOD)
# print (matrix)

ans = 0
for i in range(N):
    for j in range(N):
        ans += matrix[i][j]
        ans %= MOD

ans %= MOD
print (ans)