AT_abc455_b [ABC455B] Spiral Galaxy

There is a grid with $ H $ rows and $ W $ columns. The cell at the $ i $ -th row from the top and $ j $ -th column from the left is denoted as cell $ (i, j) $ . Each cell of the grid is colored white or black. The information of the grid is given by $ H $ strings $ S_1, S_2, \ldots, S_H $ each of length $ W $ : cell $ (i, j) $ is white if the $ j $ -th character of $ S_i $ is `.`, and black if it is `#`. Find the number of rectangular regions of the grid that are point-symmetrically colored. More formally, find the number of integer tuples $ (h_1, h_2, w_1, w_2) $ satisfying all of the following conditions: - $ 1 \leq h_1 \leq h_2 \leq H $ - $ 1 \leq w_1 \leq w_2 \leq W $ - For all integers $ i, j $ satisfying $ h_1 \leq i \leq h_2 $ and $ w_1 \leq j \leq w_2 $ , cell $ (i, j) $ and cell $ (h_1 + h_2 - i, w_1 + w_2 - j) $ have the same color.

The input is given from Standard Input in the following format: > $ H $ $ W $ $ S_1 $ $ S_2 $ $ \vdots $ $ S_H $

Output the answer.

### Sample Explanation 1  As shown in the figure above, the answer is $ 10 $ . ### Constraints - $ 1 \leq H, W \leq 10 $ - $ H $ and $ W $ are integers. - $ S_i $ is a string of length $ W $ consisting of `.` and `#`.