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: Time Limit Exceeded)

n,*a = $stdin.read.split.map(&:to_i)
sub = [[]]
a.each do |x|
	sub.concat sub.map{|y| y + [x]}
end
sub.shift
s = Array.new
sub.each do |n|
	s << n.inject(:+)
end
s.sort!

puts (s.size % 2).zero? ? s[s.size/2 - 1, 2].inject(:+) / 2.0 : s[s.size/2]