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: Accepted)

use proconio::*;

fn main() {
    input! {
        n: usize,
        s: i64,
    }

    let mut v = vec![];
    for _ in 0..n {
        input! {
            x: i64,
        }

        v.push(x);
    }
    
    let m = 998244353;
    let mut d = [0; 3001];
    let mut p = [0; 3001];
    d[0] = 1;
    for i in 0..n {
        for j in 0..s+1 {
            let iter = (s - j) as usize;
            p[iter] = d[iter] * 2;
            p[iter] %= m;
            if iter >= v[i] as usize {
                p[iter] += d[iter - v[i] as usize];
                p[iter] %= m;
            }
        }
        for j in 0..s as usize + 1 {
            d[j] = p[j];
        }
    }

    println!("{}", p[s as usize]);
}