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:

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

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.

Constraints

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

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

见题面 Input 部分

见题面 Output 部分