G - Power Pair

Score : $600$ points

Problem Statement

Given is a prime number $P$.

How many pairs of integers $(x, y)$ satisfy the following conditions?

• $0 \leq x \leq P-1$
• $0 \leq y \leq P-1$
• There exists a positive integer $n$ such that $x^n \equiv y \pmod{P}$.

Since the answer may be enormous, print it modulo $998244353$.

Constraints

• $2 \leq P \leq 10^{12}$
• $P$ is a prime number.

Input

Input is given from Standard Input in the following format:

$P$

Output

Print the answer modulo $998244353$.

3

4

Four pairs $(x, y) = (0, 0), (1, 1), (2, 1), (2, 2)$ satisfy the conditions.

11

64

998244353

329133417