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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


入力例 1

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)

入力例 2

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)

入力例 3

7 1000
1 2 3 4 3 2 1


出力例 3

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.

Sample Input 1

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)

Sample Input 2

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)

Sample Input 3

7 1000
1 2 3 4 3 2 1


Sample Output 3

225650129


Sample Input 4

5 1000000000000000000
1 1 1 1 1


Sample Output 4

307835847