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)

fn main() {
    proconio::input! {
        n: usize,
        s: usize,
        a: [usize;n],
    }
    let mut dp = vec![std::collections::BTreeMap::new();n+2];
    dp[0].insert(0,1);
    for i in 0..n {
        let a=a[i];
        for j in (0..=i).rev() {
            dp.swap(j+1,n+1);
            let mut m=dp.pop().unwrap();
            for (&k,&v) in &dp[j] {
                if k+a<=s {
                    *m.entry(k+a).or_insert(0)+=v;
                }
            }
            dp.push(m);
            dp.swap(j+1,n+1);
        }
    }
    let mut p=vec![1;n];
    for i in 1..n {
        p[i] = p[i-1]*2%MOD;
    }
    let mut t=0;
    for i in 1..=n {
        if let Some(&v)=dp[i].get(&s) {
            t = (t+v*p[n-i])%MOD;
        }
    }
    println!("{}",t);
}
const MOD: usize=998244353;