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 Haskell (GHC 8.8.3) (judge verdict: Time Limit Exceeded)

import Data.Array
import Data.Bool
import qualified Data.ByteString.Char8 as C
import Data.List

main = C.interact $ C.pack . show . sol . unfoldr (C.readInt . C.dropWhile (<'0'))

sol (n:s:as) = dp!(n,s)
  where
  ar = listArray (1,n) $ sort as
  bs = ((0,0),(n,s))
  dp = listArray bs $ f <$> range bs
  f (0,j) = bool 0 1 (j==0)
  f (i,0) = p2 i
  f (i,j) = (`mod` p) $ 2*dp!(i-1,j) + bool 0 (dp!(i-1,j')) (inRange bs (i-1,j'))
    where j' = j-ar!i

p = 998244353 :: Int

p2 :: Int -> Int
p2 x
  | x == 0    = 1
  | odd x     = 2*r `mod` p
  | otherwise = r
  where
  q = p2 (x `div` 2)
  r = q*q `mod` p