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# (.NET Core 3.1.201) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Text;
using System.Linq;

namespace contest_Csharp
{
    class Program
    {
        static long N, S;
        static List<long> A;
        static long[] kazu;

        static void Main(string[] args)
        {
            var sw = new System.IO.StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
            Console.SetOut(sw);
           
            var vals = Console.ReadLine().Split(' ')
                .Select(x => long.Parse(x)).ToList();
            long ans = 0;
            N = vals[0];
            S = vals[1];
            long mod = 998244353;
            A = Console.ReadLine().Split(' ')
                .Select(x => long.Parse(x)).ToList();
            A.Sort();
            kazu = new long[N+1];
            serch(1, S,(int)N);
            for (int i = 0; i < N+1; i++)
            {
                ans = (ans + kazu[i] * power(2, N - i, mod)) % mod;
            }



            Console.WriteLine(ans);
            Console.Out.Flush();
        }
        static void serch(long m,long goal,int limit)
        {
            int current = 0;
            while (current<limit && A[current]<=goal)
            {
                if (A[current]==goal)
                {
                    kazu[m]++;
                }
                else
                {
                    if(m<N+1)
                    serch(m + 1, goal - A[current],current);
                }
                current++;
            }
        }
        static long power(long a, long b, long modulo)
        {
            long result = 1;
            long buff = a;
            for (int i = 1; i <= b; i *= 2)
            {
                result = result * ((b / i % (2) == 1) ? buff : 1) % modulo;
                buff = (buff * buff) % modulo;
            }
            return result % modulo;
        }



    }
}