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 Lua (Lua 5.3.5) (judge verdict: Accepted)

local n,k=io.read("n","n")
local a={}
for i=1,n do
    a[i]={}
    for j=1,n do
        a[i][j]=io.read("n")
    end
end

local mod=1000000007
local function mulmatrix(a,b)
    local A=#a
    local B=#b
    local C=#b[1]
    local c={}
    for i=1,A do
        c[i]={}
        for j=1,C do
            c[i][j]=0
        end
    end
    for i=1,A do
        for k=1,B do
            for j=1,C do
                c[i][j]=(c[i][j]+a[i][k]*b[k][j])%mod
            end
        end
    end
    return c
end

local function powmatrix(a,k)
    local A=#a
    local b={}
    for i=1,A do
        b[i]={}
        for j=1,A do
            b[i][j]=i==j and 1 or 0
        end
    end
    while k>0 do
        if k&1>0 then
            b=mulmatrix(b,a)
        end
        a=mulmatrix(a,a)
        k=k>>1
    end
    return b
end

local A=powmatrix(a,k)
local total=0
for i=1,n do
    for j=1,n do
        total=(total+A[i][j])%mod
    end
end
print(total)