Contest Duration: - (local time) (100 minutes) Back to Home
Ex - Product Modulo 2 /

Time Limit: 2 sec / Memory Limit: 1024 MB

### 問題文

998244353 で割った余りを求めてください。

• すべての i(1\leq i\leq K) について、0\leq A_i \leq M-1

• \displaystyle\prod_{i=1}^{K} A_i \equiv N \pmod M

### 制約

• 1 \leq K \leq 10^9
• 0 \leq N \lt M \leq 10^{12}
• 入力は全て整数である

### 入力

K N M


### 入力例 1

2 3 6


### 出力例 1

5


### 入力例 2

10 0 2


### 出力例 2

1023


### 入力例 3

1000000000 20220326 1000000000000


### 出力例 3

561382653


998244353 で割った余りを求めてください。

Score : 600 points

### Problem Statement

Among the sequences of length K consisting of integers, A=(A_1, \ldots, A_K), how many satisfy all of the conditions below?
Find the count modulo 998244353.

• 0\leq A_i \leq M-1 for every i(1\leq i\leq K).

• \displaystyle\prod_{i=1}^{K} A_i \equiv N \pmod M.

### Constraints

• 1 \leq K \leq 10^9
• 0 \leq N \lt M \leq 10^{12}
• All values in input are integers.

### Input

Input is given from Standard Input in the following format:

K N M


### Sample Input 1

2 3 6


### Sample Output 1

5


The five sequences A satisfying the conditions are (1,3),(3,1),(3,3),(3,5),(5,3).

### Sample Input 2

10 0 2


### Sample Output 2

1023


### Sample Input 3

1000000000 20220326 1000000000000


### Sample Output 3

561382653


Be sure to find the count modulo 998244353.