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ZOJ Problem Set - 3955
Saddle Point

Time Limit: 1 Second      Memory Limit: 131072 KB

Chiaki has an n × m matrix A. Rows are numbered from 1 to n from top to bottom and columns are numbered from 1 to m from left to right. The element in the i-th row and the j-th column is Ai, j.

Let M({i1, i2, ..., is}, {j1, j2, ..., jt}) be the matrix that results from deleting row i1, i2, ..., is and column j1, j2, ..., jt of A and f({i1, i2, ..., is}, {j1, j2, ..., jt}) be the number of saddle points in matrix M({i1, i2, ..., is}, {j1, j2, ..., jt}).

Chiaki would like to find all the value of f({i1, i2, ..., is}, {j1, j2, ..., jt}). As the output may be very large ((2n - 1)(2m - 1) matrix in total), she is only interested in the value

$$\left(\sum_{\begin{array}{r} 1 \le i_1 < \dots < i_s \le n \\ 1 \le j_1 < \dots < j_t \le m \\ 0 \le s < n \\ 0 \le t < m\end{array}} f(\{i_1,i_2,\dots,i_s\},\{j_1,j_2,\dots,j_t\})\right) \bmod (10^9+7).$$

Note that a saddle point of a matrix is an element which is both the only largest element in its column and the only smallest element in its row.

Input

There are multiple test cases. The first line of input contains an integer T, indicating the number of test cases. For each test case:

The first line contains four integers n and m (1 ≤ n, m ≤ 1000) -- the number of rows and the number of columns.

Each of the next n lines contains m integer Ai, 1, Ai, 2, ..., Ai, m (1 ≤ Ai, j ≤ 106), where Ai, j is the integer in the i-th row and the j-th column.

It is guaranteed that neither the sum of all n nor the sum of all m exceeds 5000.

Output

For each test case, output an integer denoting the answer.

Sample Input

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

Sample Output

4
465

Author: LIN, Xi
Source: The 17th Zhejiang University Programming Contest Sponsored by TuSimple
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