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)

mod = 998244353

n, s = gets.to_s.split.map{|e| e.to_i }
a = gets.to_s.split.map{|e| e.to_i }.sort

# h = Hash.new(0)
h = Array.new(s+1){ [0] * (n+1) }
h[0][0] = 1
ms = 0
a.each_with_index do |t, i|
  
  ms = [ms+t, s].min 
  i.downto(0) do |y|
    ms.downto(t) do |x|
      h[x][y+1] += h[x-t][y]
    end
  end
end
# pp h
ans = 0
t = 1
n.downto(1) do |i|
  
  # p tmp = h[[s,i]]
  tmp = h[s][i] * t
  ans += tmp
  ans %= mod
  t *= 2
end

puts ans