Score : $500$ points

问题描述

整数序列展览正在进行，将各种整数序列汇聚一堂并进行评估。在此活动中，每个长度为 $N$ 的整数序列（其元素取值范围在 $1$ 到 $K$ 之间，包括两端点）都会被评估，并赋予一个介于 $1$ 和 $M$（包括两端点）之间的整数评分。

请计算并输出给定评价序列分配从 $1$ 到 $M$ 分的方案数模 $998244353$ 的结果。

这里需要注意的是，若存在某个已评估序列 $A = (A_1, A_2, \ldots, A_N)$ 在两种不同的方案中得到了不同的评分，则这两种方案被认为是不同的。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Integer Sequence Exhibition is taking place, where integer sequences are gathered in one place and evaluated. Here, every integer sequence of length $N$ consisting of integers between $1$ and $K$ (inclusive) is evaluated and given an integer score between $1$ and $M$ (inclusive).

Print the number, modulo $998244353$, of ways to give each of the evaluated sequences a score between $1$ and $M$.

Here, two ways are said to be different when there is an evaluated sequence $A = (A_1, A_2, \ldots, A_N)$ that is given different scores by the two ways.

Constraints

• $1 \leq N, K, M \leq 10^{18}$
• $N$, $K$, and $M$ are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$ $M$

Output

Print the number, modulo $998244353$, of ways to give each of the evaluated sequences a score between $1$ and $M$.

Sample Input 1

2 2 2

Sample Output 1

16

Four sequences are evaluated: $(1, 1)$, $(1, 2)$, $(2, 1)$, and $(2, 2)$. There are $16$ ways to give each of these sequences a score between $1$ and $2$, as follows.

• Give $1$ to $(1, 1)$, $1$ to $(1, 2)$, $1$ to $(2, 1)$, and $1$ to $(2, 2)$
• Give $1$ to $(1, 1)$, $1$ to $(1, 2)$, $1$ to $(2, 1)$, and $2$ to $(2, 2)$
• Give $1$ to $(1, 1)$, $1$ to $(1, 2)$, $2$ to $(2, 1)$, and $1$ to $(2, 2)$
• Give $1$ to $(1, 1)$, $1$ to $(1, 2)$, $2$ to $(2, 1)$, and $2$ to $(2, 2)$
• $\cdots$
• Give $2$ to $(1, 1)$, $2$ to $(1, 2)$, $2$ to $(2, 1)$, and $2$ to $(2, 2)$

Thus, we print $16$.

Sample Input 2

3 14 15926535

Sample Output 2

109718301

Be sure to print the count modulo $998244353$.

update @ 2024/3/10 09:59:04