Description

We are given a basketball court with $n$ horizontal division lines and $m$ vertical division lines, dividing the court into an $n \times m$ grid of regions.

The coach assigns a unique formation number (from $0$ to $nm-1$) to each region, forming a formation matrix.

The formation diversity value of a matrix is defined as the number of distinct values of the minimum missing formation number across all submatrices of the matrix.

::::info[Example]{open} If the formation numbers in a submatrix are $\{0,1,3,4\}$, the minimum missing number is $2$. If the formation numbers in a submatrix are $\{0,2\}$, the minimum missing number is $1$. ::::

Find the sum of the formation diversity values over all possible formation matrices. Since the answer can be large, output it modulo $998244353$.

Formal Statement

Given $n, m$. An $n \times m$ matrix is valid if it is a permutation of integers from $0$ to $nm-1$. Define $val$ of a matrix as the size of the set containing the $\text{mex}$ of all its submatrices. Compute the sum of $val$ over all valid matrices, modulo $998244353$.

::::success[Definition of Submatrix] A submatrix is formed by deleting a prefix of rows/columns and a suffix of rows/columns from the original matrix. ::::

Input Format

One line with two integers $n, m$.

Output Format

One integer representing the answer modulo $998244353$.

1 2

6

1 20

704442100

10 10

506259194

Hint

Explanation of Sample 1

There are two valid matrices:

1. 0 1
2. 1 0

For 0 1, the submatrices are 0 (mex = $1$), 1 (mex = $0$), 0 1 (mex = $2$). The set is $\{0,1,2\}$, so $val = 3$. The case 1 0 is symmetric. Total answer: $3 + 3 = 6$.

Data Constraints

Special Property A: $n = 1$. Special Property B: $n, m \le 15$. Special Property C: $n, m \le 80$. Special Property D: $n, m \le 400$.

For $100\%$ data: $1 \le n \times m \le 5 \times 10^6$. Please pay attention to the constant complexity of your implementation.