102284: [AtCoder]ABC228 E - Integer Sequence Fair

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 $998244353$.

问题描述

整数序列展示会正在举行，所有整数序列都集中在一个地方进行评估。在这里，每个长度为$N$的整数序列，由$1$到$K$（包括）之间的整数组成，都会被评估并给出一个在$1$到$M$（包括）之间的整数分数。

打印一种方法，使得对每个评估过的序列给出$1$到$M$之间的分数，其方式的数量对$998244353$取模。

当有两种方式时，如果存在一个被评估的序列$A = (A_1, A_2, \ldots, A_N)$，在这两种方式中被赋予不同的分数，则称这两种方式是不同的。

约束

• $1 \leq N, K, M \leq 10^{18}$
• $N$，$K$和$M$都是整数。

输入

输入从标准输入中给出，格式如下：

$N$ $K$ $M$

输出

打印一种方法，使得对每个评估过的序列给出$1$到$M$之间的分数，其方式的数量对$998244353$取模。

样例输入1

2 2 2

样例输出1

16

四个序列被评估：$(1, 1)$，$(1, 2)$，$(2, 1)$和$(2, 2)$。有$16$种方式对这些序列中的每个序列给出$1$到$2$之间的分数，如下所示。

• 将$1$赋予$(1, 1)$，$1$赋予$(1, 2)$，$1$赋予$(2, 1)$，$1$赋予$(2, 2)$
• 将$1$赋予$(1, 1)$，$1$赋予$(1, 2)$，$1$赋予$(2, 1)$，$2$赋予$(2, 2)$
• 将$1$赋予$(1, 1)$，$1$赋予$(1, 2)$，$2$赋予$(2, 1)$，$1$赋予$(2, 2)$
• 将$1$赋予$(1, 1)$，$1$赋予$(1, 2)$，$2$赋予$(2, 1)$，$2$赋予$(2, 2)$
• $\cdots$
• 将$2$赋予$(1, 1)$，$2$赋予$(1, 2)$，$2$赋予$(2, 1)$，$2$赋予$(2, 2)$

因此，我们打印$16$。

样例输入2

3 14 15926535

样例输出2

109718301

确保打印的计数对$998244353$取模。