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 Ruby (2.7.1) (judge verdict: Time Limit Exceeded)

M=998244353
n,s,*as=$<.read.split.map &:to_i
t=1
P2s=(0...n).map{|i| tc=t; t=t*2%M; tc }
as.sort!{|a| -a }
ss=[]
t=0
(n-1).downto(0){|i|
  ss[i]=t
  t+=as[i]
}
w={ 0i=>[1,0] }
aks={}
as.each_with_index{|a,i|
  lim=s-ss[i]
  ks=w.keys
  ks.each{|k|
    x,y=k.rect
    y2=y+a
    if y2<=s
      x+=1
      aks[x+y2*1i]=true if y2==s
      r=w[k]
      (w[x+y2*1i]||=[0,0])[1]+=r[0]
    end
    w.delete(k) if y<lim
  }
  w.values.each{|r|
    r[0]+=r[1]
    r[1]=0
  }
}
puts aks.keys.reduce(0){|s,k|
  v=w[k][0]
  (s+v*P2s[n-k.real])%M
}