Problem:
Score : 700 points

Problem Statement

You are given N integers A_1, A_2, ..., A_N.

Consider the sums of all non-empty subsequences of A. There are 2^N - 1 such sums, an odd number.

Let the list of these sums in non-decreasing order be S_1, S_2, ..., S_{2^N - 1}.

Find the median of this list, S_{2^{N-1}}.

Constraints

1 \leq N \leq 2000

1 \leq A_i \leq 2000

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the median of the sorted list of the sums of all non-empty subsequences of A.

Sample Input 1

3
1 2 1

Sample Output 1

2

In this case, S = (1, 1, 2, 2, 3, 3, 4). Its median is S_4 = 2.

Sample Input 2

1
58

Sample Output 2

58

In this case, S = (58).

Solution in Ruby (2.3.3) (judge verdict: Wrong Answer)

n = gets.to_i
a = gets.split.map &:to_i

if n==1
  p a[0]
  exit
elsif n==2
  p a.max
  exit
end

dp = (n+1).times.map{Array.new(n+1, 0)}
dp[0][0] = 1

n.times{|i|
  n.times{|j|
    break if dp[i][j]==0
    dp[i+1][j] += dp[i][j]
    dp[i+1][j+1] += dp[i][j]+a[i]
  }
}

sum =  dp[n].inject(:+)-(n*(n+1)/2)-1
puts sum/(2**n)