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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;//リストの使用
using System.Collections.Generic;
class Program
{
  static long n = long.Parse(Console.ReadLine());
  static long[] nums = Array.ConvertAll(Console.ReadLine().Split(' '),long.Parse);
  static long numsSum = 0;//数列の総和
  static long answer = long.MaxValue;

	static void Main()
	{
      for(long i = 0; i < n; i++) numsSum += nums[i];
      
      long partitionLeft = 0;//仕切りの位置。数字が左に1個だけある状態が0
      long partitionRight = 2;
      long sumLeftLeft = nums[0];
      long sumLeft = nums[1];
      long sumRight = nums[2];
      long sumRightRight = numsSum-nums[0]-nums[1]-nums[2];

      for(long partitionMiddle = 1; partitionMiddle < n-2; partitionMiddle++)//中央の仕切りの位置
      {
      //Console.WriteLine(partitionLeft + " " + partitionMiddle + " " + partitionRight);
        while(true)//左の仕切りを右側へ移動
        {
          if(partitionLeft < partitionMiddle - 1 && 
            Math.Abs(sumLeftLeft + 2*nums[partitionLeft+1] - sumLeft)
              < Math.Abs(sumLeftLeft - sumLeft))
              {
                sumLeftLeft += nums[partitionLeft + 1];
                sumLeft -= nums[partitionLeft + 1];
                partitionLeft++;
              }else
              {
                break;
              }
        }

        while(true)//右の仕切りを右側へ移動
        {
          if(partitionRight < n-2 && 
            Math.Abs(sumRight + 2*nums[partitionRight+1] - sumRightRight)
              < Math.Abs(sumRight - sumRightRight))
              {
                sumRight += nums[partitionRight + 1];
                sumRightRight -= nums[partitionRight + 1];
                partitionRight++;
              }else
              {
                break;
              }
        }
        
        long[] memoLong = new long[6];
        memoLong[0] = Math.Abs(sumLeftLeft - sumLeft);
        memoLong[1] = Math.Abs(sumLeftLeft - sumRight);
        memoLong[2] = Math.Abs(sumLeftLeft - sumRightRight);
        memoLong[3] = Math.Abs(sumLeft - sumRight);
        memoLong[4] = Math.Abs(sumLeft - sumRightRight);
        memoLong[5] = Math.Abs(sumRight - sumRightRight);
        answer = Math.Min(answer, memoLong.Max());//最小値の更新
        //Console.WriteLine(sumLeftLeft + " " + sumLeft + " " + sumRight + " " + sumRightRight);

        sumLeft += nums[partitionMiddle+1];
        sumRight -= nums[partitionMiddle+1];
        if(partitionRight == partitionMiddle+1)
        {
          partitionRight++;
          sumRight += nums[partitionMiddle+2];
          sumRightRight -= nums[partitionMiddle+2];
        }
      }

      Console.WriteLine(answer);
	}
  
}