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 Java (OpenJDK 11.0.6) (judge verdict: Accepted)

import java.util.*;
import java.io.*;

public class Main {
    static long mod = (long) 1e9 + 7;

    public static void main(String[] args) throws IOException {
        FastScanner in = new FastScanner(System.in);
        PrintWriter out = new PrintWriter(System.out);
        int n = in.nextInt();
        long k = in.nextLong();
        long[][] base = new long[n][n];
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                base[i][j] = in.nextInt();
            }
        }
        //print(base);
        //System.out.println();
        long[][] pow = expo(base, k);
        //print(pow);
        long ans = 0;
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                ans = (ans + pow[i][j]) % mod;
            }
        }
        out.println(ans);
        out.close();
    }

    static long[][] expo(long[][] base, long n) {
        long[][] ret = new long[base.length][base[0].length];
        for (int i = 0; i < base.length; i++) ret[i][i] = 1;

        while (n > 0) {
            if (n % 2 == 1) {
                ret = multiply(ret, base);
            }
            n /= 2;
            base = multiply(base, base);
        }
        return ret;
    }

    static long[][] multiply(long[][] a, long[][] b) {
        long[][] c = new long[a.length][b[0].length];
        for (int i = 0; i < a.length; i++) {
            for (int j = 0; j < b.length; j++) {
                for (int k = 0; k < b[0].length; k++) {
                    c[i][k] = (c[i][k] + a[i][j] * b[j][k]) % mod;
                }
            }
        }
        return c;
    }

    static void print(long[][] a) {
        for (int i = 0; i < a.length; i++) {
            for (int j = 0; j < a[0].length; j++) {
                System.out.print(a[i][j] + " ");
            }
            System.out.println();
        }
    }


    static class FastScanner {
        BufferedReader br;
        StringTokenizer st;
        public FastScanner(InputStream i) {
            br = new BufferedReader(new InputStreamReader(i));
            st = new StringTokenizer("");
        }
        public String next() throws IOException {
            if (st.hasMoreTokens()) {
                return st.nextToken();
            }
            else
                st = new StringTokenizer(br.readLine());
            return next();
        }
        public int nextInt() throws IOException {
            return Integer.parseInt(next());
        }
        public double nextDouble() throws IOException {
            return Double.parseDouble(next());
        }
        public long nextLong() throws IOException {
            return Long.parseLong(next());
        }
        public String nextLine() throws IOException {
            if (!st.hasMoreTokens()) {
                st = new StringTokenizer(br.readLine());
                return nextLine();
            }

            String ret = "";
            while (st.hasMoreTokens()) {
                ret += st.nextToken();
            }
            return ret;
        }
        public int[] nextIntArr(int size) throws IOException {
            int[] arr = new int[size];
            for (int i = 0; i < arr.length; i++) {
                arr[i] = nextInt();
            }
            return arr;
        }
        public long[] nextLongArr(int size) throws IOException {
            long[] arr = new long[size];
            for (int i = 0; i < arr.length; i++) {
                arr[i] = nextLong();
            }
            return arr;
        }
        public double[] nextDoubleArr(int size) throws IOException {
            double[] arr = new double[size];
            for (int i = 0; i < arr.length; i++) {
                arr[i] = nextDouble();
            }
            return arr;
        }
    }
}