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: Wrong Answer)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Text;
using System.Linq;
using static MyIO;


public class D
{
	public static void Main() => (new Solver()).Solve();
}


public class Solver
{
	public void Solve()
	{
		int N = GetInt();
		long[] A = GetLongs(N);

		long[] csum = new long[N + 1];
		for(int i = 0; i < N; i++)
			csum[i + 1] = csum[i] + A[i];

		long ans = long.MaxValue / 2;
		for(int i = 2; i < N - 1; i++)
		{
			int leftSpl = Math.Max(1, meguruBinarySearch(csum, 1, i));
			long P = csum[leftSpl];
			long Q = csum[i] - csum[leftSpl];
			
			int rightSpl = Math.Max(i + 1, meguruBinarySearch(csum, i + 1, N));
			long R = csum[rightSpl] - csum[i];
			long S = csum[N] - csum[rightSpl];

			long max = Math.Max(P, Math.Max(Q, Math.Max(R, S)));
			long min = Math.Min(P, Math.Min(Q, Math.Min(R, S)));
			ans = Math.Min(ans, max - min);
		}

		Console.WriteLine(ans);
	}

	private int meguruBinarySearch(long[] csum, int first, int last)
	{
		long sum = csum[last] - csum[first - 1];
		long minDiff = long.MaxValue / 2;
		int ok = first - 1;
		int ng = last;
		while(Math.Abs(ng - ok) > 1)
		{
			int mid = (ng + ok) / 2;
			long diff = Math.Abs(sum - 2 * (csum[mid] - csum[first - 1]));
			if(diff < minDiff)
			{
				ok = mid;
				minDiff = diff;
			}
			else
				ng = mid;
		}
		return ok;
	}
}


public static class MyIO
{
	private static string[] args = null;
	private static int num = -1;
	private static int used = -1;

	private static string getArg()
	{
		if(used == num)
		{
			args = Console.ReadLine().Split(' ');
			num = args.Length;
			used = 0;
		}
		return args[used++];
	}

	public static int GetInt() => int.Parse(getArg());
	public static long GetLong() => long.Parse(getArg());
	public static double GetDouble() => double.Parse(getArg());
	public static string GetString() => getArg();
	public static char GetChar() => getArg()[0];
	public static int[] GetInts(int N) => Enumerable.Range(0, N).Select(_ => GetInt()).ToArray();
	public static long[] GetLongs(int N) => Enumerable.Range(0, N).Select(_ => GetLong()).ToArray();
	public static double[] GetDoubles(int N) => Enumerable.Range(0, N).Select(_ => GetDouble()).ToArray();
	public static string[] GetStrings(int N) => Enumerable.Range(0, N).Select(_ => GetString()).ToArray();
}