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: Time Limit Exceeded)

def nibutan(as, left_num, n)
  len  = n-1
  left = 0
  right = len
  while left+1 != right
    middle = (left+right)/2
    first = ((as[middle]-left_num)-(as[len] -as[middle])).abs
    second =((as[middle+1]-left_num)-(as[len] -as[middle+1])).abs
    if first > second
      left = middle
    else
      right = middle
    end
  end
  if ((as[left]-left_num)-(as[len] -as[left])).abs < ((as[right]-left_num)-(as[len] -as[right])).abs
    left
  else
    right
  end
end

n = gets.to_i
as = gets.split.map(&:to_i)
sum = [0]
n.times{|i|
  sum << as[i]+sum[i]
}
sum = sum[1..-1]
result = 100000000000

2.upto(n-2){|i|
  left_list = sum[0..i-1].clone
  right_list = sum[i..-1].clone
  left_i = nibutan(left_list, 0, i)
  right_i = nibutan(right_list, sum[i-1], n-i)

  #p [as[0..left_i], as[left_i+1..i-1], as[i..i+right_i], as[i+right_i+1..-1]]
  res = [sum[left_i], sum[i-1]-sum[left_i], sum[i+right_i]-sum[i-1], sum[-1]-sum[i+right_i]]
  result = (res.max - res.min) < result ?  (res.max - res.min) : result
}
puts result