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 C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
using ll = int_fast64_t;
const int mod1 = 1e9 + 7;
const int mod2 = 998244353;
const int inf = (1 << 30) - 1;
const ll infll = (1LL << 61) - 1;
#define fast() ios::sync_with_stdio(false), cin.tie(nullptr)
#define digit(N) cout << fixed << setprecision((N))
#pragma GCC target("avx2")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")

int N, S, dp[3030][3030], a[3030];

int main()
{
	fast();

	cin >> N >> S;

	for (int i = 0; i < N; i++)
	{
		cin >> a[i];
	}

	dp[0][0] = 1;

	for (int i = 0; i < N; i++)
	{
		for (int j = 0; j <= S; j++)
		{
			if (j + a[i] <= S)
				dp[i + 1][j + a[i]] = (dp[i][j] + dp[i + 1][j + a[i]]) % mod2;
			dp[i + 1][j] = ((2 * dp[i][j]) % mod2 + dp[i + 1][j]) % mod2;
		}
	}
	cout << dp[N][S] << "\n";
}