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)

#[allow(unused_macros)]
macro_rules! input {
    ( source = $ s : expr , $ ( $ r : tt ) * ) => { let mut iter = $ s . split_whitespace ( ) ; let mut next = || { iter . next ( ) . unwrap ( ) } ; input_inner ! { next , $ ( $ r ) * } };
    ( $ ( $ r : tt ) * ) => { let stdin = std :: io :: stdin ( ) ; let mut bytes = std :: io :: Read :: bytes ( std :: io :: BufReader :: new ( stdin . lock ( ) ) ) ; let mut next = move || -> String { bytes . by_ref ( ) . map ( | r | r . unwrap ( ) as char ) . skip_while ( | c | c . is_whitespace ( ) ) . take_while ( | c |! c . is_whitespace ( ) ) . collect ( ) } ; input_inner ! { next , $ ( $ r ) * } };
}
#[allow(unused_macros)]
macro_rules! input_inner {
    ( $ next : expr ) => { };
    ( $ next : expr , ) => { };
    ( $ next : expr , $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let $ var = read_value ! ( $ next , $ t ) ; input_inner ! { $ next $ ( $ r ) * } };
}
#[allow(unused_macros)]
macro_rules! read_value {
    ( $ next : expr , ( $ ( $ t : tt ) ,* ) ) => { ( $ ( read_value ! ( $ next , $ t ) ) ,* ) };
    ( $ next : expr , [ $ t : tt ; $ len : expr ] ) => { ( 0 ..$ len ) . map ( | _ | read_value ! ( $ next , $ t ) ) . collect ::< Vec < _ >> ( ) };
    ( $ next : expr , chars ) => { read_value ! ( $ next , String ) . chars ( ) . collect ::< Vec < char >> ( ) };
    ( $ next : expr , usize1 ) => { read_value ! ( $ next , usize ) - 1 };
    ( $ next : expr , $ t : ty ) => { $ next ( ) . parse ::<$ t > ( ) . expect ( "Parse error" ) };
}

#[allow(unused_macros)]
macro_rules! vec_md {
    ( $ init : expr ; $ ( $ dim : expr ) ;* ) => { vec_md ! ( $ ( $ dim ) ;* => $ init ) };
    ( $ head : expr ; $ ( $ tail : expr ) ;* => $ init : expr ) => { vec ! [ vec_md ! ( $ ( $ tail ) ;* => $ init ) ; $ head ] };
    ( $ head : expr => $ init : expr ) => { vec ! [ $ init ; $ head ] };
}

#[allow(dead_code)]
pub const M: u64 = 1_000_000_007;

fn main() {
    input! {
        n: usize, k: u64,
        a: [[u64; n]; n]
    }

    let init = vec![1u64;n];
    let a_k = matrix_pow(&a, k);
    let mut ret = 0u64;
    for i in 0..n {
        let row = &a_k[i];
        ret += dot(row, &init);
        ret %= M;
    }
    println!("{}", ret);
}

#[allow(dead_code)]
fn dot(a: &[u64], b: &[u64]) -> u64 {
    let mut c = 0;
    for i in 0..a.len() {
        c += a[i] * b[i];
        c %= M;
    }
    c
}

#[allow(dead_code)]
fn matrix_mul(a: &[Vec<u64>], b: &[Vec<u64>]) -> Vec<Vec<u64>> {
    let mut c = vec![vec![0; b[0].len()]; a.len()];
    for i in 0..a.len() {
        for k in 0..b.len() {
            for j in 0..b[0].len() {
                c[i][j] = c[i][j] + a[i][k] * b[k][j] % M;
                c[i][j] %= M;
            }
        }
    }
    c
}
#[allow(dead_code)]
fn matrix_pow(a_: &[Vec<u64>], mut n: u64) -> Vec<Vec<u64>> {
    let mut a: Vec<Vec<u64>> = a_.to_vec();
    let mut b = vec![vec![0; a.len()]; a.len()];
    for i in 0..a.len() {
        b[i][i] = 1;
    }
    while n > 0 {
        if n & 1 == 1 {
            b = matrix_mul(&b, &a);
        }
        a = matrix_mul(&a, &a);
        n >>= 1;
    }
    b
}