This is a stretch problem.

There is an ~n \times m~ grid of cells. Cell ~(r, c)~ has a positive weight ~w_{r, c}~. Let ~W = \sum_{r, c} w_{r, c}~.

Two cells ~A~ and ~B~ are sampled independently, with replacement. The probability of choosing cell ~(r, c)~ on a single sample is ~w_{r, c} / W~.

Let ~R~ be the smallest axis-aligned rectangle of grid cells that contains both sampled cells, and let ~A~ be the number of cells inside ~R~ (the area of ~R~ counted in cells). If ~A~ and ~B~ are the same cell, then ~R~ is that single cell and the area is ~1~.

Output the expected value of ~A~.

It can be shown that the answer can be expressed as a rational number ~P/Q~ in lowest terms with ~Q~ coprime to ~10^9 + 7~. Output ~P \cdot Q^{-1} \bmod (10^9 + 7)~.

Input

The first line contains two integers ~n~ and ~m~.

Each of the next ~n~ lines contains ~m~ integers. The ~c~-th number on the ~r~-th of these lines is ~w_{r, c}~.

Rows and columns are indexed from ~1~.

Output

Print a single integer: the expected number of cells in the bounding rectangle, modulo ~10^9 + 7~.

Constraints

• ~1 \le n, m~
• ~n \cdot m \le 2 \cdot 10^5~
• ~1 \le w_{r, c} \le 10^6~

Example 1

Input

2 2
1 1
1 1

Output

250000004

Explanation

All ~16~ ordered pairs of cells are equally likely. The expected area is ~9/4~, and ~9 \cdot 4^{-1} \equiv 250000004 \pmod{10^9 + 7}~.

Example 2

Input

1 1
5

Output

1

Explanation

Both samples are the only cell, so the rectangle always has area ~1~.

Example 3

Input

1 3
1 1 1

Output

888888897

Explanation

The expected area is ~17/9~, and ~17 \cdot 9^{-1} \equiv 888888897 \pmod{10^9 + 7}~.