Problem Statement

Find the number, modulo 998244353, of sequences X that satisfy all of the following conditions.

• Every term in X is a positive odd number.
• The sum of the terms in X is S.
• The prefix sums of X contain none of $A_1, \dots, A_N.$ Formally, if we define $Y_i = X_1 + \dots + X_i$ for each i, then $Y_i \neq A_j$ holds for all integers i and j such that $1 \leq i \leq |X|$ and $1 \leq j \leq N$.

Input

Input is given from Standard Input in the following format:
N S
$A_1 \ldots A_N$

Output

Print the answer.

Constraints

• $1 \leq N \leq 10^5$
• $1 \leq A_1 \lt A_2 \lt \dots \lt A_N \lt S \leq 10^{18}$
• All values in input are integers.

Sample Input 1

3 7
2 4 5

Sample Output 1

3

The following three sequences satisfy the conditions.

• (1, 5, 1)
• (3, 3, 1)
• (7)

Sample Input 2

5 60
10 20 30 40 50

Sample Output 2

37634180

Sample Input 3

10 1000000000000000000
1 2 4 8 16 32 64 128 256 512

Sample Output 3

75326268

见题面 Input 部分

见题面 Output 部分