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)

lines = $stdin.read
array = lines.split("\n")
N = array[0].to_i
A = array[1].split(" ").map(&:to_i)

subset = []

1.upto(N) do |take_n|
  A.combination(take_n) do |sub|
    subset << sub.inject(&:+)
  end
end

subset.sort!

if subset.length % 2 == 0
  l = subset.length/2
  r = l - 1
  puts [subset[l], subset[r]].inject(&:+) / 2
else
  puts subset[subset.length/2]
end