Score : $600$ points

问题陈述

有一个 $1 \times N$ 的网格，从左到右编号为 $1,2,\dots,N$。

高桥准备了 $C$ 种颜色的颜料，并用这 $C$ 种颜色中的任意一种给每个小方格涂色。
然后，在任意连续的 $K$ 个小方格中，最多有两种颜色。
形式化地讲，对于满足 $1 \le i \le N-K+1$ 的所有整数 $i$，在第 $i,i+1,\dots,i+K-1$ 个小方格中最多有两种颜色。

高桥有多少种不同的涂色方式？
由于这个数字可能非常大，请计算它对 $998244353$ 取模的结果。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There is a grid with $1 \times N$ squares, numbered $1,2,\dots,N$ from left to right.

Takahashi prepared paints of $C$ colors and painted each square in one of the $C$ colors.
Then, there were at most two colors in any consecutive $K$ squares.
Formally, for every integer $i$ such that $1 \le i \le N-K+1$, there were at most two colors in squares $i,i+1,\dots,i+K-1$.

In how many ways could Takahashi paint the squares?
Since this number can be enormous, find it modulo $998244353$.

Constraints

• All values in the input are integers.
• $2 \le K \le N \le 10^6$
• $1 \le C \le 10^9$

Input

The input is given from Standard Input in the following format:

$N$ $K$ $C$

Output

Print the answer as an integer.

Sample Input 1

3 3 3

Sample Output 1

21

In this input, we have a $1 \times 3$ grid.
Among the $27$ ways to paint the squares, there are $6$ ways to paint all squares in different colors, and the remaining $21$ ways are such that there are at most two colors in any consecutive three squares.

Sample Input 2

10 5 2

Sample Output 2

1024

Since $C=2$, no matter how the squares are painted, there are at most two colors in any consecutive $K$ squares.

Sample Input 3

998 244 353

Sample Output 3

952364159

Print the number modulo $998244353$.

update @ 2024/3/10 11:45:24