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: Accepted)

class EqualCutter
  attr_accessor :arr, :culm
  attr_accessor :left, :center, :right
  def initialize(arr)
    @arr = arr
    @left = 0
    @center = 1
    @right = 2
    @culm = [arr[0]]
    (1..arr.size() - 1).each do |i|
      @culm[i] = @culm[i - 1] + arr[i]
    end
  end

  def p
    @culm[@left]
  end

  def nextP
    @culm[@left+1]
  end

  def q
    @culm[@center] - @culm[@left]
  end

  def nextQ
    @culm[@center] - @culm[@left+1]
  end

  def r
    @culm[@right] - @culm[@center]
  end

  def nextR
    @culm[@right+1] - @culm[@center]
  end

  def s
    @culm[@arr.size() - 1] - @culm[@right]
  end

  def nextS
    @culm[@arr.size() - 1] - @culm[@right+1]
  end

  def leftIncrementable
    (p - q).abs > (nextP - nextQ).abs && @left+1 < @center
  end

  def rightIncrementable
    (r - s).abs > (nextR - nextS).abs && @right+1 < @arr.size() - 1
  end

  def diff
    [p, q, r, s].max - [p, q, r, s].min
  end

  def print
    puts("left: #{@left}, center: #{@center}, right: #{@right}")
    puts("p: #{p}, q: #{q}, r: #{r}, s: #{s}")
  end
end

n = gets().chomp().to_i()
arr = gets().chomp().split().map!(&:to_i)

diff = 1000000001

ec = EqualCutter.new(arr)

(1..n - 3).each do |center|
  ec.center = center
  while(ec.leftIncrementable) do
    ec.left += 1
  end
  
  while(ec.rightIncrementable) do
    ec.right += 1
  end

  diff = ec.diff if diff > ec.diff
end

puts diff