Grid Completion

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