Home

Contest: Task: Related: TaskG TaskI

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.

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