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 Rust (1.42.0) (judge verdict: Wrong Answer)

use proconio::input;

#[allow(unused_mut)]
#[allow(non_snake_case)]
fn main() {
    input! {
        N: usize,
        S: usize,
        A: [usize; N]
    }

    let mut dp: Vec<Vec<u64>> = vec![vec![0; S + 1]; N + 1];

    let m = 998244353;

    // 和が0を初期化
    for i in 1..=N {
        dp[i][0] = dp[i - 1][0] * 2 + 1;
    }
    for i in 1..=N {
        for j in 1..=S {
            if j == A[i - 1] {
                dp[i][j] = (dp[i][j] + 1) % m;
            }

            // 入れないグループ
            dp[i][j] = (dp[i][j] + dp[i - 1][j] * 2) % m;

            // A[i]を追加するパターン
            if A[i - 1] <= j && j - A[i - 1] <= S {
                dp[i][j] = (dp[i][j] + dp[i - 1][j - A[i - 1]]) % m;
            }
        }
    }

    println!("{}", dp[N][S]);
}