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 $1 \\leq j \\leq N$.

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.

Input

Input is given from Standard Input in the following format:

$N$ $S$ $A_1$ $\\ldots$ $A_N$

Output

Print the answer.

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