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 Rust (1.15.1) (judge verdict: Accepted)

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}

type mat = Vec<Vec<i64>>;

fn mat_zeros(n: usize, m: usize) -> mat {
    vec![vec![0i64; m]; n]
}

const big_int: i64 = 1000_000_000 + 7;

fn matmul(a: &mat, b: &mat) -> mat {
    let mut c = mat_zeros(a.len(), b[0].len());
    for i in 0..a.len() {
        for k in 0..b.len() {
            for j in 0..b[0].len() {
                c[i][j] += a[i][k] * b[k][j];
                c[i][j] %= big_int;
            }
        }
    }
    c
}

fn matpow(a: &mat, n: u64) -> mat {
    let mut b = mat_zeros(a.len(), a.len());
    for i in 0..a.len() {
        b[i][i] = 1;
    }
    let mut n = n;
    let mut a: mat = a.clone();
    while n > 0 {
        if n & 1 == 1 {
            b = matmul(&a, &b);
        }
        a = matmul(&a, &a);
        n >>= 1;
    }
    b
}

fn main() {
    let v = read_vec::<usize>();
    let (n, k) = (v[0], v[1]);
    let mut a = mat_zeros(n, n);
    for i in 0..n {
        a[i] = read_vec();
    }
    a = matpow(&a, k as u64);
    let mut ans = 0;
    for i in 0..n {
        for j in 0..n {
            ans += a[i][j];
            ans %= big_int;
        }
    }
    println!("{}", ans);
}