Your task is to create an $n \times n$ grid whose each row and column has exactly one A and B. Some of the characters have already been placed. In how many ways can you complete the grid?

Input

The first input line has an integer $n$: the size of the grid.

After this, there are $n$ lines that describe the grid. Each line has $n$ characters: . means an empty square, and A and B show the characters already placed.

You can assume that every row and column has at most one A and B.

Output

Print one integer: the number of ways modulo $10^9+7$.

Constraints

• $2 \le n \le 500$

Example

Input:

5
.....
..AB.
.....
B....
...A.

Output:

16