Problem:
Score : 600 points

Problem Statement

Snuke has an integer sequence A of length N.

He will make three cuts in A and divide it into four (non-empty) contiguous subsequences B, C, D and E.
The positions of the cuts can be freely chosen.

Let P,Q,R,S be the sums of the elements in B,C,D,E, respectively.
Snuke is happier when the absolute difference of the maximum and the minimum among P,Q,R,S is smaller.
Find the minimum possible absolute difference of the maximum and the minimum among P,Q,R,S.

Constraints

4 \leq N \leq 2 \times 10^5

1 \leq A_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Find the minimum possible absolute difference of the maximum and the minimum among P,Q,R,S.

Sample Input 1

5
3 2 4 1 2

Sample Output 1

2

If we divide A as B,C,D,E=(3),(2),(4),(1,2), then P=3,Q=2,R=4,S=1+2=3.
Here, the maximum and the minimum among P,Q,R,S are 4 and 2, with the absolute difference of 2.
We cannot make the absolute difference of the maximum and the minimum less than 2, so the answer is 2.

Sample Input 2

10
10 71 84 33 6 47 23 25 52 64

Sample Output 2

36

Sample Input 3

7
1 2 3 1000000000 4 5 6

Sample Output 3

999999994

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

n = gets.to_i
as = gets.split.map(&:to_i)

h = {0 => as[0]}
as.each.with_index do |a,i|
  next if i == 0
  h[i] = h[i-1] + a
end

ans = Float::INFINITY
(2..n-2).each do |i|
  left = as[0, i]
  left_sum = h[i-1]
  left_i = Float::INFINITY
  l_tmp = Float::INFINITY
  (1..left.size-1).bsearch do |l|
    lsum = left[0, l].reduce(0){|a,e|a+=e}
    tmp = ((left_sum / 2) - lsum).abs
    l_tmp = [l_tmp, tmp].min
    if l_tmp == tmp
      left_i = lsum
    end
    tmp
  end
  right = as[i, n-i]
  right_sum = h[n-1] - h[i-1]
  right_i = Float::INFINITY
  r_tmp = Float::INFINITY
  (1..right.size-1).bsearch do |l|
    rsum = right[0, l].reduce(0){|a,e|a+=e}
    tmp = ((right_sum / 2) - rsum).abs
    r_tmp = [r_tmp, tmp].min
    if r_tmp == tmp
      right_i = rsum
    end
    tmp
  end

  array = [left_i, left_sum - left_i, right_i, right_sum - right_i]
  ans = [ans, array.max - array.min].min
end

puts ans