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
def extended_gcd(a, b)
  last_remainder, remainder = a.abs, b.abs
  x, last_x, y, last_y = 0, 1, 1, 0
  while remainder != 0
    last_remainder, (quotient, remainder) = remainder, last_remainder.divmod(remainder)
    x, last_x = last_x - quotient*x, x
    y, last_y = last_y - quotient*y, y
  end
  return last_remainder, last_x * (a < 0 ? -1 : 1)
end
def invmod(e)
  g, x = extended_gcd(e, M)
  x % M
end

inv2=invmod(2)
n,s,*as=$<.read.split.map &:to_i
p2n=1
n.times{ p2n=p2n*2%M }
as.sort!{|a| -a }
w={ 0=>[p2n,0] }
as.each_with_index{|a,i|
  ks=w.keys
  ks.each{|k|
    k2=k+a
    next if k2>s
    r=w[k]
    r2=(w[k2]||=[0,0])
    r2[1]=(r2[1]+r[0]*inv2)%M
  }
  w.values.each{|r|
    r[0]=(r[0]+r[1])%M
    r[1]=0
  }
}
puts (w[s]||[0,0])[0]