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

namespace TOPSIC
{
	public class Program
	{
		public static void Main()
		{
			int n = int.Parse(ReadLine());
			var list = ReadLine().Split().Select(x => int.Parse(x))
				.ToList();

			// 累積和を使う
			var sum = new List<double>();
			sum.Add(list[0]);
			for (int i = 1; i < n; i++)
			{
				sum.Add(sum[i - 1] + list[i]);
			}

			var ans = double.MaxValue;
			var list2 = new double[4];

			// 中央の区切りは全探索
			// 中央の区切りが移動した場合は、計算済みの左区切りと右区切りを利用する
			int left = 0;
			int right = 2;
			for (int i = 1; i < n - 2; i++)
			{
				// 左区間の区切り
				while (left < i &&
					 // sum[left + 1] - (sum[i] - sum[left + 1]) = 2 * sum[left + 1] - sum[i]
					 // sum[left] - (sum[i] - sum[left]) = sum[left] - sum[i] + sum[left] = 2 * sum[left] - sum[i]
					 // 境界を求める。一つ先が減っているなら、次へ移動する。
					 Math.Abs(2 * sum[left + 1] - sum[i]) < Math.Abs(2 * sum[left] - sum[i]))
				{
					left++;
				}

				// 右区間の区切り
				while (right < n &&
					// (sum[right] - sum[i]) - (sum[n-1] - sum[right]) = sum[right] - sum[i] - sum[n-1] - sum[right] = 2 * sum[right] - sum[i] - sum[n-1]
					Math.Abs(2 * sum[right + 1] - sum[i] - sum[n - 1]) < Math.Abs(2 * sum[right] - sum[i] - sum[n - 1]))
				{
					right++;
				}

				list2[0] = sum[left];
				list2[1] = sum[i] - sum[left];
				list2[2] = sum[right] - sum[i];
				list2[3] = sum[n - 1] - sum[right];
				ans = Math.Min(ans, list2.Max() - list2.Min());
			}

			WriteLine(ans);
		}
	}
}