Limestone

AtCoder

IDarc199_d

Time2000ms

Memory256MB

Difficulty—

You are given positive integers $H,W$. There is an $H \times W$ matrix $A=(A_{i,j})\ (1\leq i\leq H,1\leq j\leq W)$, where initially all elements are $0$. You can perform the following two types of operations on this matrix in any order and any number of times: * Choose integers $r,c\ (1\leq r\leq H,1\leq c\leq W)$. Set all of $A_{r,1},A_{r,2},\ldots,A_{r,c}$ to $1$. * Choose integers $r,c\ (1\leq r\leq H,1\leq c\leq W)$. Set all of $A_{1,c},A_{2,c},\ldots,A_{r,c}$ to $1$. Find the sum of $\displaystyle \sum_{i=1}^H\sum_{j=1}^W A_{i,j}$ over all matrices that can be obtained by repeating the above operations, modulo $998244353$. ## Constraints * $1\leq H,W$ * $HW\leq 2\times 10^5$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $H$ $W$ [samples]

Samples

Input #1

2 2

Output #1

29

There are $14$ matrices that can be obtained by repeating the operations:

$\begin{pmatrix}0&0 \\ 0&0\end{pmatrix} \begin{pmatrix}1&0 \\ 0&0\end{pmatrix} \begin{pmatrix}0&1 \\ 0&0\end{pmatrix} \begin{pmatrix}1&1 \\ 0&0\end{pmatrix} \begin{pmatrix}0&0 \\ 1&0\end{pmatrix} \begin{pmatrix}1&0 \\ 1&0\end{pmatrix} \begin{pmatrix}0&1 \\ 1&0\end{pmatrix} \begin{pmatrix}1&1 \\ 1&0\end{pmatrix} \begin{pmatrix}0&1 \\ 0&1\end{pmatrix} \begin{pmatrix}1&1 \\ 0&1\end{pmatrix} \begin{pmatrix}0&0 \\ 1&1\end{pmatrix} \begin{pmatrix}1&0 \\ 1&1\end{pmatrix} \begin{pmatrix}0&1 \\ 1&1\end{pmatrix} \begin{pmatrix}1&1 \\ 1&1\end{pmatrix}$

The answer is $0+1+1+2+1+2+2+3+2+3+2+3+3+4=29$.
$\begin{pmatrix}0&0 \\ 0&1\end{pmatrix}$ and $\begin{pmatrix}1&0 \\ 0&1\end{pmatrix} $ cannot be obtained by any sequence of operations.

Input #2

1 10

Output #2

Input #3

123 456

Output #3

428623282

API Response (JSON)

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    "name": "Limestone",
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      "content": "You are given positive integers $H,W$. There is an $H \\times W$ matrix $A=(A_{i,j})\\ (1\\leq i\\leq H,1\\leq j\\leq W)$, where initially all elements are $0$. You can perform the following two types of op",
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      "memory_limit": 262144
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    "difficulty": "None",
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    "sync_url": null,
    "sign": "arc199_d"
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      "statement_type": "Markdown",
      "content": "You are given positive integers $H,W$.\nThere is an $H \\times W$ matrix $A=(A_{i,j})\\ (1\\leq i\\leq H,1\\leq j\\leq W)$, where initially all elements are $0$.\nYou can perform the following two types of op...",
      "is_translate": false,
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