Digit-only subrectangles

There are $H$ rows and $W$ columns of square cells. Each cell has either a digit or an asterisk (‘*’). The cell at the $i$-th row from the top and the $j$-th column from the left is denoted by $(i,j)$.

In this problem we consider subrectangles, each of which is the set of cells which forms a rectangle. More precisely, a set of cells $S$ is a subrectangle if there are four integers $t$, $b$, $l$ and $r$ such that $1 \le t \le b \le H$, $1 \le l \le r \le W$ and $S = \{(i,j) \mid t \le i \le b \wedge l \le j \le r \}$. A subrectangle is digit-only if every cell in the subrectangle has a digit. The score of a digit-only subrectangle is defined as the square of the sum of digits in cells in the subrectangle.

Your task is to calculate the sum of scores of all digit-only subrectangles. Since the answer may be large, output it modulo $998\,244\,353$.

The input consists of a single test case of the following format.

$H$ $W$

$A_{1,1}A_{1,2}\cdots A_{1,W}$

$A_{2,1}A_{2,2}\cdots A_{2,W}$

$\vdots$

$A_{H,1}A_{H,2}\cdots A_{H,W}$

The first line consists of two integers $H$ and $W$, which satisfy $1 \le H \le 2\,000$ and $1 \le W \le 2\,000$. Each of the following $H$ lines consists of $W$ characters. Here, $A_{i,j}$ is the character in the cell $(i,j)$, and it is either a digit between $0$ and $9$, inclusive, or an asterisk (‘*’). It is guaranteed that there is at least one digit-only subrectangle.

Output in a line the sum of scores of all digit-only subrectangles modulo $998\,244\,353$.

In Sample Input 1, there are five digit-only subrectangles as illustrated below. The sum of their scores is $4^2 + 4^2 + 9^2 + (4 + 4)^2 + (4 + 9)^2 = 346$.

Figure F.1. Digit-only subrectangles in Sample Input 1