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Ex - Odd Steps

Score : $600$ points

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.

3 7
2 4 5

3

The following three sequences satisfy the conditions.

• $(1, 5, 1)$
• $(3, 3, 1)$
• $(7)$

5 60
10 20 30 40 50

37634180

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

75326268