Rook Path

AtCoder

IDabc232_e

Time2000ms

Memory256MB

Difficulty—

There is a $H \times W$\-square grid with $H$ horizontal rows and $W$ vertical columns. Let $(i, j)$ denote the square at the $i$\-th row from the top and $j$\-th column from the left. The grid has a rook, initially on $(x_1, y_1)$. Takahashi will do the following operation $K$ times. * Move the rook to a square that shares the row or column with the square currently occupied by the rook. Here, it must move to a square different from the current one. How many ways are there to do the $K$ operations so that the rook will be on $(x_2, y_2)$ in the end? Since the answer can be enormous, find it modulo $998244353$. ## Constraints * $2 \leq H, W \leq 10^9$ * $1 \leq K \leq 10^6$ * $1 \leq x_1, x_2 \leq H$ * $1 \leq y_1, y_2 \leq W$ ## Input Input is given from Standard Input in the following format: $H$ $W$ $K$ $x_1$ $y_1$ $x_2$ $y_2$ [samples]

Samples

Input #1

2 2 2
1 2 2 1

Output #1

2

We have the following two ways.

*   First, move the rook from $(1, 2)$ to $(1, 1)$. Second, move it from $(1, 1)$ to $(2, 1)$.
*   First, move the rook from $(1, 2)$ to $(2, 2)$. Second, move it from $(2, 2)$ to $(2, 1)$.

Input #2

1000000000 1000000000 1000000
1000000000 1000000000 1000000000 1000000000

Output #2

24922282

Be sure to find the count modulo $998244353$.

Input #3

3 3 3
1 3 3 3

Output #3

API Response (JSON)

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  "problem": {
    "name": "Rook Path",
    "description": {
      "content": "There is a $H \\times W$\\-square grid with $H$ horizontal rows and $W$ vertical columns. Let $(i, j)$ denote the square at the $i$\\-th row from the top and $j$\\-th column from the left. The grid has a ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc232_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a $H \\times W$\\-square grid with $H$ horizontal rows and $W$ vertical columns. Let $(i, j)$ denote the square at the $i$\\-th row from the top and $j$\\-th column from the left.\nThe grid has a ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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