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: Runtime Error)

using System;
using System.Collections.Generic;
using System.Linq;
using static System.Console;

namespace AtCoder
{
	public class Program
	{
		/// <summary>
		/// ABC 102
		/// </summary>
		public static void Main()
		{
			int n = int.Parse(ReadLine());
			var list = ReadLine().Split().Select(x => int.Parse(x))
				.ToList();

			// 高速化のために、累積和を先に求めておく
			var sum = new List<double>();
			sum.Add(list[0]);
			for(int i = 1; i<n; i++)
			{
				sum.Add(sum[i - 1] + list[i]);
			}

			// 目的は左右の合計の差の最小値を探すこと。
			double min = double.MaxValue;
			var list2 = new double[4];

			// 中央の区切り
			for (int i = 1; i<n; i++)
			{
				// 左の区切り
				var leftMin = double.MaxValue;
				var leftTotal = sum[i];
				for (int j = 0; j < i; j++)
				{
					var leftA = sum[j];
					var leftB = leftTotal - leftA;
					var leftD = Math.Abs(leftA - leftB);
					if (leftD < leftMin)
					{
						leftMin = leftD;
						list2[0] = leftA;
						list2[1] = leftB;
					}

					// 右の区切りを決める
					var rightMin = double.MaxValue;
					var rightSum = 0D;
					var rightTotal = list.Skip(i+1).Sum();
					for (int k = i + 1; k < n; k++)
					{
						rightSum += list[k];

						var rightA = rightSum;
						var rightB = rightTotal - rightA;
						var rightD = Math.Abs(rightA - rightB);
						if (rightD < rightMin)
						{
							rightMin = rightD;
							list2[2] = rightA;
							list2[3] = rightB;
						}

						var m = Math.Abs(list2.Max() - list2.Min());
						if (m < min)
						{
							min = m;
						}
					}
				}
			}

			WriteLine(min);
		}
	}
}