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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Threading.Tasks;
using static System.Console;
using static System.Math;
class Z { static void Main() => new K(); }
class K
{
	int F => int.Parse(Str);
	long FL => long.Parse(Str);
	int[] G => Strs.Select(int.Parse).ToArray();
	long[] GL => Strs.Select(long.Parse).ToArray();
	double[] GR => Strs.Select(double.Parse).ToArray();
	string Str => ReadLine();
	string[] Strs => Str.Split(new char[] { ' ', '\t' }, StringSplitOptions.RemoveEmptyEntries);
	public const int MOD = 1000000007;
	public const long INFL = 4011686018427387913L;
	public K()
	{
		SetOut(new StreamWriter(OpenStandardOutput()) { AutoFlush = false });
		Solve();
		Out.Flush();
	}
	void Solve()
	{
		var I = GL;
		var N = (int)I[0];
		var K = I[1];
		var m = new Matrix(N);
		for (var i = 0; i < N; i++)
		{
			var I2 = G;
			for (var j = 0; j < N; j++) m.a[i, j] = I2[j];
		}
		m = m.Pow((ulong)K);
		var ans = 0L;
		for (var i = 0; i < N; i++) for (var j = 0; j < N; j++) ans += m.a[i, j];
		WriteLine(ans % MOD);
	}
}
class Matrix
{
	const int MOD = K.MOD;
	public int Size { get; protected set; }
	public long[,] a;
	public Matrix(int r) { Size = r; a = new long[r, r]; }
	public Matrix(long[,] c) : this(c.GetLength(0), (i, j) => c[i, j]) { }
	public Matrix(int r, Func<int, int, long> f) : this(r) { for (var i = 0; i < r; i++) for (var j = 0; j < r; j++) a[i, j] = f(i, j) % MOD; }
	public static Matrix operator *(Matrix p, Matrix q)
		=> new Matrix(p.Size, (i, j) => { var sum = 0L; for (var k = 0; k < p.Size; k++) sum = (sum + p.a[i, k] * q.a[k, j] % MOD) % MOD; return sum; });
	public static Matrix One(int size) => new Matrix(size, (i, j) => i == j ? 1 : 0);
	public Matrix Pow(ulong n)
	{
		var p = One(Size);
		var x = this;
		while (n > 0)
		{
			if ((n & 1) == 1) p = p * x;
			n >>= 1;
			x = x * x;
		}
		return p;
	}
}