Problem:
Score : 600 points

Problem Statement

Given are a sequence of N positive integers A_1, A_2, \ldots, A_N and another positive integer S.

For a non-empty subset T of the set \{1, 2, \ldots , N \}, let us define f(T) as follows:

f(T) is the number of different non-empty subsets \{x_1, x_2, \ldots , x_k \} of T such that A_{x_1}+A_{x_2}+\cdots +A_{x_k} = S.

Find the sum of f(T) over all 2^N-1 subsets T of \{1, 2, \ldots , N \}. Since the sum can be enormous, print it modulo 998244353.

Constraints

All values in input are integers.

1 \leq N \leq 3000

1 \leq S \leq 3000

1 \leq A_i \leq 3000

Input

Input is given from Standard Input in the following format:

N S
A_1 A_2 ... A_N

Output

Print the sum of f(T) modulo 998244353.

Sample Input 1

3 4
2 2 4

Sample Output 1

6

For each T, the value of f(T) is shown below. The sum of these values is 6.

f(\{1\}) = 0

f(\{2\}) = 0

f(\{3\}) = 1 (One subset \{3\} satisfies the condition.)

f(\{1, 2\}) = 1 (\{1, 2\})

f(\{2, 3\}) = 1 (\{3\})

f(\{1, 3\}) = 1 (\{3\})

f(\{1, 2, 3\}) = 2 (\{1, 2\}, \{3\})

Sample Input 2

5 8
9 9 9 9 9

Sample Output 2

0

Sample Input 3

10 10
3 1 4 1 5 9 2 6 5 3

Sample Output 3

3296

Solution in PyPy3 (7.3.0) (judge verdict: Wrong Answer)

MOD = 998244353


def main():
    from itertools import accumulate
    import sys
    sys.setrecursionlimit(10**6)
    N, S = (int(i) for i in input().split())
    A = [int(i) for i in input().split()]
    A.sort(reverse=True)
    acc = list(accumulate([0] + A))
    depth = set()

    def dfs(i, v):
        if i == N:
            if v == S:
                depth.add(i)
                return 1
            else:
                return 0
        if v == S:
            depth.add(i)
            return 2**(N-i)
        if S < v or v + (acc[N] - acc[i]) < S:
            return 0
        res = 0
        res += dfs(i+1, v + A[i]) + dfs(i+1, v)
        return res % MOD

    ans = dfs(0, 0) % MOD
    print(ans)


if __name__ == '__main__':
    main()