Charlie has a grid G with N rows and M columns. The rows are numbered 0 through $N-1$ from top to bottom. The columns are numbered 0 through $M-1$ from left to right.

Each cell of the grid G contains a positive integer in the range $[1, N\times M]$. Each of these numbers occurs in the grid exactly once.

In Charlie's eyes, the most beautiful of all grids is the sorted grid $G_S$ : A grid of size $N \times M$ whose list of elements in row major order is the ordered sequence $[1,2,3,...,N\times M]$. For example with $N=2$ and $M=3$, $G_S = \begin{bmatrix} 1 & 2 & 3\newline 4 & 5 & 6 \end{bmatrix}$

Denote $F(G)$ = Number of pairs $(i,j)$ such that $G[i][j] = G_S[i][j], 0 ≤ i < N, 0 ≤ j < M$

For example:
$F\Bigg(\begin{bmatrix} 1 & 2 & 3\newline 4 & 5 & 6 \end{bmatrix}\Bigg) = 6,$ $F\Bigg(\begin{bmatrix} 3 & 2 & 1\newline 5 & 4 & 6 \end{bmatrix}\Bigg) = 2$

Charlie can also modify the grid by applying the following two operations any number of times:

• Swap any two rows
• Swap any two columns

Let S denote the set of distinct grids G that Charlie can generate from the input grid G.

You need to print the sum $\displaystyle \sum\limits_{G'\in S} F(G')^2$
Since the result can be large, print it modulo $10^9+7$

Input Format:

The first line will contain two integers $N, M$.
The next N lines will contain M integers each. The $i^{th}$ line describes the numbers in the $i^{th}$ row of G.
The input grid numbers are guaranteed to be a permutation of $[1,2,3,..,N\times M]$.

Output Format:

Output the desired sum modulo $10^9+7$ in a single line.

Constraints

There are 100 files. Your solution will get a score equal to the number of files you pass. Some files may have different bounds, as detailed below. For all subtasks:
$2 ≤ N, M$

Subtask 1 (36 files):
$N, M ≤ 3$

Subtask 2 (27 files):
$N, M ≤ 7$

Subtask 3 (18 files):
$N, M ≤ 50$

Subtask 4 (9 files):
$N, M ≤ 300$

Subtask 5 (10 files):
$N, M ≤ 1500$