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 PyPy3 (7.3.0) (judge verdict: Time Limit Exceeded)

n,s=map(int,input().split())
N=[i+1 for i in range(n)]
list1=[]
pattern=[]
A=list(map(int,input().split()))

ft=0  
dict={}
for i in range(n+1):
    dict[i]=[]
for i in range(1,2**n):
    list2=[]
    for j in range(n):
        if (i >> j) &1 == 1:
            list2.append(N[j])
    pattern=[]
    for x in range(1,2**len(list2)):
        pt=[]
        for y in range(len(list2)):
            if (x>>y)&1==1:
                pt.append(list2[y])
        pattern.append(pt)
   # print(pattern)

    for p in pattern:
        sum=0
        if p in dict[len(p)]:
            ft+=1
        else:
            for x in p:
                sum+=A[x-1]
            if sum==s:
                dict[len(p)].append(p)
                ft+=1
                ft=ft%998244353
print(ft%998244353)