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.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Runtime.CompilerServices;
using System.Text;
using System.Threading.Tasks;

namespace AtCorder
{
	public class Program
	{
		static void Main(string[] args)
		{
			var cin = new Scanner2();
			int n = cin.Int();
			long[] a = cin.ArrayLong(n);

			long sum = a.Sum();
			var ruiL = new long[n + 1];
			var ruiR = new long[n + 1];
			for (int i = 0; i < n; i++) {
				ruiL[i + 1] = a[i] + ruiL[i];
				ruiR[i + 1] = a[n - 1 - i] + ruiR[i];
			}

			long ans = sum;
			for (int i = 1; i < n - 2; i++) {
				long leftSum = ruiL[i + 1];
				long rightSum = sum - leftSum;
				int li = BinarySearchIntL(0, i, mid => ruiL[mid + 1] <= leftSum / 2);
				int ri = BinarySearchIntL(
					i + 1, n - i, mid => ruiL[mid + 1] - ruiL[i + 1] <= rightSum / 2);

				for (int j = 0; j < 2; j++) {
					for (int k = 0; k < 2; k++) {
						long l1 = j == 0 || li == i ? ruiL[li + 1] : ruiL[li + 2];
						long l2 = leftSum - l1;
						long r1 = k == 0 || ri == n - 2 
							? ruiL[ri + 1] - ruiL[i + 1]
							: ruiL[ri + 2] - ruiL[i + 1];
						long r2 = rightSum - r1;
						
						long min = Math.Min(l1, Math.Min(l2, Math.Min(r1, r2)));
						long max = Math.Max(l1, Math.Max(l2, Math.Max(r1, r2)));
						ans = Math.Min(ans, max - min);
					}
				}
			}

			Console.WriteLine(ans);
		}

		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static int BinarySearchIntL(int ok, int ng, Func<int, bool> check)
		{
			while (ng - ok > 1) {
				int mid = (ok + ng) / 2;
				if (check(mid)) {
					ok = mid;
				} else {
					ng = mid;
				}
			}

			return ok;
		}
	}

	public static class Helper
	{
		public static void UpdateMin32<T>(T array, int index, int value) where T : IList<int>
			=> array[index] = Math.Min(array[index], value);
		public static void UpdateMax32<T>(T array, int index, int value) where T : IList<int>
			=> array[index] = Math.Max(array[index], value);
		public static void UpdateMin64<T>(T array, int index, long value) where T : IList<long>
			=> array[index] = Math.Min(array[index], value);
		public static void UpdateMax64<T>(T array, int index, long value) where T : IList<long>
			=> array[index] = Math.Max(array[index], value);

		public static T[] Array<T>(int n, Func<int, T> init)
			=> Enumerable.Range(0, n).Select(x => init(x)).ToArray();
		public static List<T> List<T>(int n, Func<int, T> init)
			=> Enumerable.Range(0, n).Select(x => init(x)).ToList();

		private static readonly int[] delta4_ = { 1, 0, -1, 0, 1 };
		public static void DoAt4(int i, int j, int imax, int jmax, Action<int, int> action)
		{
			for (int n = 0; n < 4; n++) {
				int ii = i + delta4_[n];
				int jj = j + delta4_[n + 1];
				if ((uint)ii < (uint)imax && (uint)jj < (uint)jmax) {
					action(ii, jj);
				}
			}
		}
	}

	public class Scanner2
	{
		private readonly char[] delimiter_ = new char[] { ' ' };
		private readonly string filePath_;
		private readonly Func<string> reader_;
		private string[] buf_;
		private int index_;

		public Scanner2(string file = "")
		{
			if (string.IsNullOrWhiteSpace(file)) {
				reader_ = Console.ReadLine;
			} else {
				filePath_ = file;
				var fs = new StreamReader(file);
				reader_ = fs.ReadLine;
			}
			buf_ = new string[0];
			index_ = 0;
		}

		public string Next()
		{
			if (index_ < buf_.Length) {
				return buf_[index_++];
			}

			string st = reader_();
			while (st == "") {
				st = reader_();
			}

			buf_ = st.Split(delimiter_, StringSplitOptions.RemoveEmptyEntries);
			if (buf_.Length == 0) {
				return Next();
			}

			index_ = 0;
			return buf_[index_++];
		}

		public int Int() => int.Parse(Next());
		public long Long() => long.Parse(Next());
		public double Double() => double.Parse(Next());

		public int[] ArrayInt(int N, int add = 0)
		{
			int[] Array = new int[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Int() + add;
			}
			return Array;
		}

		public long[] ArrayLong(int N, long add = 0)
		{
			long[] Array = new long[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Long() + add;
			}
			return Array;
		}

		public double[] ArrayDouble(int N, double add = 0)
		{
			double[] Array = new double[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Double() + add;
			}
			return Array;
		}

		public void Save(string text)
		{
			if (string.IsNullOrWhiteSpace(filePath_)) {
				return;
			}

			File.WriteAllText(filePath_ + "_output.txt", text);
		}
	}
}