G - At Most 2 Colors

Score : $600$ points

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.

3 3 3

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.

10 5 2

1024

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

998 244 353

952364159

Print the number modulo $998244353$.