P9359 [ICPC 2022 Xi'an R] Cells Coloring

You are given an $n \times m$ grid. Some of the cells are obstacles, the others are empty. Choose a non-negative integer $k$ and color all empty cells with $k+1$ colors $0, 1, 2, \ldots k$. You can not color two cells in the same row or same column with the same **non-zero** color. You are given two non-negative integers $c$ and $d$. For a coloring plan, define $z$ as the number of the cells with color $0$. Define the cost of the plan is $ck+dz$. Find the minimum cost.

The first line contains four integers $n$, $m$ ($1\leq n, m\leq 250$), $c$ and $d$ ($0\leq c, d\leq 10 ^ 9$). The $i$-th line of the next $n$ lines contains a string of $m$ characters. The $j$-th character is `*` if the cell in the $i$-th row and the $j$-th column is an obstacle. The $j$-th character is `.` if the cell in the $i$-th row and the $j$-th column is empty.

Output a line with a single number, representing the answer.

**Source**: The 2022 ICPC Asia Xi'an Regional Contest Problem B. **Author**: csy2005.