Contest Duration: - (local time) (120 minutes) Back to Home
F - Growth Rate /

Time Limit: 4 sec / Memory Limit: 1024 MB

### 問題文

• 1\leq X_i\leq M (1\leq i\leq N+1)
• A_iX_i\leq X_{i+1} (1\leq i\leq N)

### 制約

• 1\leq N\leq 1000
• 1\leq M\leq 10^{18}
• 1\leq A_i\leq 10^9
• \prod_{i=1}^N A_i \leq M

### 入力

N M
A_1 A_2 \ldots A_N

2 10
2 3

### 出力例 1

7

• (1, 2, 6), (1,2,7), (1,2,8), (1,2,9), (1,2,10), (1,3,9), (1,3,10)

2 10
3 2

### 出力例 2

9

• (1, 3, 6), (1, 3, 7), (1, 3, 8), (1, 3, 9), (1, 3, 10), (1, 4, 8), (1, 4, 9), (1, 4, 10), (1, 5, 10)

7 1000
1 2 3 4 3 2 1

225650129

### 入力例 4

5 1000000000000000000
1 1 1 1 1

### 出力例 4

307835847

Score : 1000 points

### Problem Statement

Given is a positive integer M and a sequence of N integers: A = (A_1,A_2,\ldots,A_N). Find the number, modulo 998244353, of sequences of N+1 integers, X = (X_1,X_2, \ldots, X_{N+1}), satisfying all of the following conditions:

• 1\leq X_i\leq M (1\leq i\leq N+1)
• A_iX_i\leq X_{i+1} (1\leq i\leq N)

### Constraints

• 1\leq N\leq 1000
• 1\leq M\leq 10^{18}
• 1\leq A_i\leq 10^9
• \prod_{i=1}^N A_i \leq M

### Input

Input is given from Standard Input in the following format:

N M
A_1 A_2 \ldots A_N

### Output

Print the number of integer sequences satisfying the conditions, modulo 998244353.

2 10
2 3

### Sample Output 1

7

Seven sequences below satisfy the conditions.

• (1, 2, 6), (1,2,7), (1,2,8), (1,2,9), (1,2,10), (1,3,9), (1,3,10)

2 10
3 2

### Sample Output 2

9

Nine sequences below satisfy the conditions.

• (1, 3, 6), (1, 3, 7), (1, 3, 8), (1, 3, 9), (1, 3, 10), (1, 4, 8), (1, 4, 9), (1, 4, 10), (1, 5, 10)

7 1000
1 2 3 4 3 2 1

225650129

### Sample Input 4

5 1000000000000000000
1 1 1 1 1

307835847