"too many constructive problems... can we have a math problem instead??"
– nobody

You are given an ~N \times M~ grid containing each of the integers from ~0~ to ~NM-1~ exactly once. Denote the integer in the ~i^{th}~ row and ~j^{th}~ column as ~a_{i,j}~.

For any grid of integers, define its set of achievable MEX values to be all integers ~k~ for which there exists a subgrid of the grid whose MEX is equal to ~k~.

Determine the number of ~N \times M~ grids containing each of the integers from ~0~ to ~NM-1~ exactly once which have the same set of achievable MEX values as the given grid, modulo ~10^9+7~.

Note: MEX means minimum excluded value. The MEX of a set of numbers ~S~ refers to the smallest non-negative integer which is not contained in ~S~.

Constraints

~1 \le N,M \le 300~
~0 \le a_{i,j} \le NM-1~

The given ~N \times M~ grid contains all integers from ~0~ to ~NM-1~ exactly once.

Subtask 1 [20%]

~N = 1~

Subtask 2 [40%]

~1 \le N,M \le 50~

Subtask 3 [40%]

No additional constraints.

Input Specification

The first line contains two integers, ~N~ and ~M~, the number of rows and the number of columns in the grid, respectively.

The next ~N~ lines each contain ~M~ integers, where the ~j^{th}~ integer on the ~i^{th}~ line is ~a_{i,j}~, describing the given grid.

Output Specification

Output one line containing one integer, the number of ~N \times M~ grids containing each of the integers from ~0~ to ~NM-1~ exactly once which have the same set of achievable MEX values as the given grid, modulo ~10^9+7~.

Sample Input 1

2 2
0 2
3 1

Sample Output 1

8

Explanation for Sample 1

The set of achievable MEX values of the given grid is ~\{0,1,4\}~. A MEX of ~0~ is achievable by taking any subgrid which does not contain ~0~ (e.g. the subgrid consisting only of the value ~1~), a MEX of ~1~ is achievable by taking the subgrid consisting only of the value ~0~, and a MEX of ~4~ is achievable by taking the subgrid consisting of the entire grid.

There are ~8~ such ~2\times 2~ grids with the same set of achievable MEX values ~\{0,1,4\}~, including the given grid. They are listed as follows:

~ \begin{bmatrix} 0 & 2 \\ 3 & 1 \end{bmatrix} \begin{bmatrix} 0 & 3 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 0 \end{bmatrix} \begin{bmatrix} 1 & 3 \\ 2 & 0 \end{bmatrix} ~

~ \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 3 & 0 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ 0 & 2 \end{bmatrix} ~

Sample Input 2

3 4
8 10 11 9
7 6 3 4
5 0 2 1

Sample Output 2

261888

Explanation for Sample 2

The set of achievable MEX values of the given grid is ~\{0, 1, 3, 5, 8, 12\}~. Let ~(r_1,c_1,r_2,c_2)~ denote the subgrid spanning rows ~r_1~ to ~r_2~ and columns ~c_1~ to ~c_2~, numbered from ~1~ to ~N~ and ~1~ to ~M~, respectively. The following is a list of the achievable MEX values followed by one of its possible corresponding subgrids.

• ~0~ : ~(1,1,1,1)~
• ~1~ : ~(3,4,3,4)~
• ~3~ : ~(3,1,3,4)~
• ~5~ : ~(2,2,3,4)~
• ~8~ : ~(2,1,3,4)~
• ~12~ : ~(1,1,3,4)~

It can be shown that no other MEX values are achievable. It can also be shown that there are exactly ~261888~ grids with ~3~ rows and ~4~ columns containing all of the numbers from ~0~ to ~11~ exactly once which have the set of achievable MEX values ~\{0, 1, 3, 5, 8, 12\}~, including the given grid.

Sample Input 3

1 20
0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1

Sample Output 3

321822771

Explanation for Sample 3

Remember to output the answer modulo ~10^9+7~.