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 LuaJIT (2.0.4) (judge verdict: Accepted)

local mfl = math.floor
local mod = 1000000007
local function badd(x, y) return (x + y) % mod end
local function bmul(x, y)
  local x1, y1 = mfl(x / 31623), mfl(y / 31623)
  local x0, y0 = x - x1 * 31623, y - y1 * 31623
  return (x1 * y1 * 14122 + (x1 * y0 + x0 * y1) * 31623 + x0 * y0) % mod
end
local function lltonumber(str)
  local ret = 0LL
  for i = 1, #str do
    ret = ret * 10LL + str:sub(i, i):byte() - 48
  end
  return ret
end

local s = io.read()
local n, k = s:match("(%d+) (%d+)")
n = tonumber(n)
k = lltonumber(k)
local t = {}
for i = 1, n do
  t[i] = {}
  for j = 1, n do
    t[i][j] = io.read("*n")
  end
end
local cpy = {}
for i = 1, n do
  cpy[i] = {}
  for j = 1, n do
    cpy[i][j] = 0
  end
end
local function update()
  for i = 1, n do
    for j = 1, n do
      cpy[i][j] = 0
    end
  end
  for i = 1, n do
    for j = 1, n do
      for k = 1, n do
        cpy[i][k] = badd(cpy[i][k], bmul(t[i][j], t[j][k]))
      end
    end
  end
  for i = 1, n do
    for j = 1, n do
      t[i][j] = cpy[i][j]
    end
  end
end
local src, dst = {}, {}
for i = 1, n do
  src[i] = 1
  dst[i] = 0
end
while 0LL < k do
  if k % 2LL == 1LL then
    for i = 1, n do
      dst[i] = 0
    end
    for i = 1, n do
      for j = 1, n do
        dst[j] = badd(dst[j], bmul(t[i][j], src[i]))
      end
    end
    for i = 1, n do
      src[i] = dst[i]
    end
  end
  update()
  k = k / 2LL
end
local ret = 0
for i = 1, n do
  ret = badd(ret, src[i])
end
print(ret)