Digit-only subrectangles

The grid has $H$ rows and $W$ columns of square cells. Each cell holds either a digit or an asterisk (‘*’). Let $(i,j)$ denote the cell in row $i$ from the top and column $j$ from the left.

A subrectangle is a set of cells that forms a rectangle. Precisely, a set $S$ is a subrectangle if there are integers $t$, $b$, $l$, $r$ with $1 \le t \le b \le H$, $1 \le l \le r \le W$, 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 it holds a digit. The score of a digit-only subrectangle is the square of the sum of its digits.

Compute the sum of scores of all digit-only subrectangles, modulo $998244353$.

The input is given in 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 contains $H$ and $W$ ($1 \le H \le 2000$, $1 \le W \le 2000$). Each of the next $H$ lines contains $W$ characters, where $A_{i,j}$ is the character in cell $(i,j)$: a digit from 0 to 9 or an asterisk (‘*’). It is guaranteed that at least one digit-only subrectangle exists.

Print the sum of scores of all digit-only subrectangles modulo $998244353$ on one line.

The first test case has five digit-only subrectangles, shown in the figure below. Their scores sum to $4^2 + 4^2 + 9^2 + (4 + 4)^2 + (4 + 9)^2 = 346$.

Figure F.1. Digit-only subrectangles in the first test case