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E - Rook Path

Score : $500$ points

Problem Statement

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:

$H$ $W$ $K$

$x_1$ $y_1$ $x_2$ $y_2$

Output

Print the number of ways to do the $K$ operations so that the rook will be on $(x_2, y_2)$ in the end, modulo $998244353$.

2 2 2
1 2 2 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 $(2, 1)$.

1000000000 1000000000 1000000
1000000000 1000000000 1000000000 1000000000

24922282

Be sure to find the count modulo $998244353$.

3 3 3
1 3 3 3

9