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: Time Limit Exceeded)

use proconio::input;

static MODC: u64 = 998244353;

fn powm(xx:u64, yy:u32) -> u64 {
    let mut x = xx;
    let mut y = yy;
    let mut result = 1;
    while y > 0 {
        if y & 1 != 0 {
            result *= x;
            result %= MODC;
        }
        x *= x;
        x %= MODC;
        y >>= 1;
    }
    result
}

fn main() {
    input! {
        n: u32, s: u32,
        mut a: [u32; n],
    };
    
    let mut stk: Vec<(u32,u32,u32)> = vec![(n-1,n,0)];
    let mut ans: u64 = 0;
    while !stk.is_empty() {
        let (index,count,sum) = stk.pop().unwrap();
        if index > 0 {
            stk.push((index-1,count,sum));
        }
        let tt = sum + a[index as usize];
        if tt == s {
            ans += powm(2,count-1);
            ans %= MODC;
        }
        else if tt < s && index > 0 {
            stk.push((index-1,count-1,tt));
        }
    }
    println!("{}", ans);
}