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)

require 'set'; require 'prime'; require 'pp'
INF=Float::INFINITY; MOD=998244353

n,s = gets.chomp.split.map(&:to_i)
a = gets.chomp.split.map(&:to_i)

@h = Hash.new(0)
a.each do |e|
  @h[e] += 1
end
# p @h

@ans = 0

@n = n
def calc(current)
  anum = 1
  nokori = @n - current
  @ans = (@ans + anum * 2.pow(nokori, MOD)) % MOD
  # p [current, anum, nokori, @ans]
end

def combination_sum2(cans, target)
  @cans = cans
  @target = target

  @cans.sort!
  @cans.select{|can| can == target}.each do |can|
    calc(1)
  end
  @cans.select!{|can| can < target}
  @size = cans.size

  solve(0, 0, 0)
end

def solve(index, current, current_sum)
  # p [index, current, current_sum]
  index.upto(@size - 1) do |i|
    nsum = current_sum + @cans[i]
    if nsum == @target
      current += 1
      calc(current)
      current -= 1
    elsif nsum < @target
      current += 1
      solve(i + 1, current, nsum)
      current -= 1
    else
      break
    end
  end
end

combination_sum2(a, s)

puts @ans