Japanese "Knight's Tour"

AtCoder

IDarc202_b

Time2000ms

Memory256MB

Difficulty—

There is a shogi (Japanese chess) board with $H$ rows and $W$ columns, and one knight piece. The square at the $i$\-th row from the top and $j$\-th column from the left of the board is represented as $(i-1, j-1)$. (Note that the values are shifted by $1$.) This board is a torus, meaning the top and bottom are connected, and the left and right are connected. A knight at $(i, j)$ can move to $((i-2) \bmod H, (j-1) \bmod W)$ or $((i-2) \bmod H, (j+1) \bmod W)$. (Note that this is different from the knight in Standard chess.) Moving the knight on the board to satisfy the following condition is called a **tour**. * Initially, place the knight at $(0, 0)$. Then, move the knight $H \times W$ times so that the knight visits every square exactly once. Finally, the knight returns to $(0, 0)$. Find the number of tours modulo $998244353$. Two tours are considered different if and only if there exists an integer $i$ $(1 \leq i \leq H\times W)$ such that the destination of the $i$\-th move is different. ## Constraints * $3 \leq H \leq 2 \times 10^5$ * $3 \leq W \leq 2 \times 10^5$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $H$ $W$ [samples]

Samples

Input #1

3 3

Output #1

6

For example, the movement $(0,0) \to (1,1) \to (2,2) \to (0,1) \to (1,2) \to (2,0) \to (0,2) \to (1,0) \to (2,1) \to (0,0)$ satisfies the conditions as a tour.  
There are six tours in total.

Input #2

123 45

Output #2

993644157

Input #3

6789 200000

Output #3

152401277

API Response (JSON)

{
  "problem": {
    "name": "Japanese \"Knight's Tour\"",
    "description": {
      "content": "There is a shogi (Japanese chess) board with $H$ rows and $W$ columns, and one knight piece. The square at the $i$\\-th row from the top and $j$\\-th column from the left of the board is represented as ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc202_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a shogi (Japanese chess) board with $H$ rows and $W$ columns, and one knight piece. The square at the $i$\\-th row from the top and $j$\\-th column from the left of the board is represented as ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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