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.IO;

namespace Contest
{
    class Scanner
    {
        public Scanner()
        {
            _stream = new StreamReader(Console.OpenStandardInput());
            _pos = 0;
            _line = new string[0];
            _separator = ' ';
        }

        private char _separator;
        private StreamReader _stream;
        private int _pos;
        private string[] _line;

        #region get a element

        public string Next()
        {
            if (_pos >= _line.Length)
            {
                _line = _stream.ReadLine().Split(_separator);
                _pos = 0;
            }

            return _line[_pos++];
        }

        public int NextInt()
        {
            return int.Parse(Next());
        }

        public long NextLong()
        {
            return long.Parse(Next());
        }

        public double NextDouble()
        {
            return double.Parse(Next());
        }

        #endregion

        #region convert array

        private int[] ToIntArray(string[] array)
        {
            var result = new int[array.Length];
            for (int i = 0; i < array.Length; i++)
            {
                result[i] = int.Parse(array[i]);
            }

            return result;
        }

        private long[] ToLongArray(string[] array)
        {
            var result = new long[array.Length];
            for (int i = 0; i < array.Length; i++)
            {
                result[i] = long.Parse(array[i]);
            }

            return result;
        }

        private double[] ToDoubleArray(string[] array)
        {
            var result = new double[array.Length];
            for (int i = 0; i < array.Length; i++)
            {
                result[i] = double.Parse(array[i]);
            }

            return result;
        }

        #endregion

        #region get row elements

        #region get array

        public string[] Array()
        {
            if (_pos >= _line.Length)
                _line = _stream.ReadLine().Split(_separator);

            _pos = _line.Length;
            return _line;
        }

        public int[] IntArray()
        {
            return ToIntArray(Array());
        }

        public long[] LongArray()
        {
            return ToLongArray(Array());
        }

        public double[] DoubleArray()
        {
            return ToDoubleArray(Array());
        }

        #endregion

        #region get 2~4 elements

        public void GetRow(out string a, out string b)
        {
            a = Next();
            b = Next();
        }

        public void GetRow(out string a, out string b, out string c)
        {
            a = Next();
            b = Next();
            c = Next();
        }

        public void GetRow(out string a, out string b, out string c, out string d)
        {
            a = Next();
            b = Next();
            c = Next();
            d = Next();
        }

        public void GetIntRow(out int a, out int b)
        {
            a = NextInt();
            b = NextInt();
        }

        public void GetIntRow(out int a, out int b, out int c)
        {
            a = NextInt();
            b = NextInt();
            c = NextInt();
        }

        public void GetIntRow(out int a, out int b, out int c, out int d)
        {
            a = NextInt();
            b = NextInt();
            c = NextInt();
            d = NextInt();
        }

        public void GetLongRow(out long a, out long b)
        {
            a = NextLong();
            b = NextLong();
        }

        public void GetLongRow(out long a, out long b, out long c)
        {
            a = NextLong();
            b = NextLong();
            c = NextLong();
        }

        public void GetLongRow(out long a, out long b, out long c, out long d)
        {
            a = NextLong();
            b = NextLong();
            c = NextLong();
            d = NextLong();
        }

        public void GetDoubleRow(out double a, out double b)
        {
            a = NextDouble();
            b = NextDouble();
        }

        public void GetDoubleRow(out double a, out double b, out double c)
        {
            a = NextDouble();
            b = NextDouble();
            c = NextDouble();
        }

        public void GetDoubleRow(out double a, out double b, out double c, out double d)
        {
            a = NextDouble();
            b = NextDouble();
            c = NextDouble();
            d = NextDouble();
        }

        #endregion

        #endregion

        #region get 2~4 column elements

        public void GetColumn(int n, out string[] a)
        {
            a = new string[n];
            for (int i = 0; i < n; i++)
            {
                a[i] = Next();
            }
        }

        public void GetColumn(int n, out string[] a, out string[] b)
        {
            a = new string[n];
            b = new string[n];
            for (int i = 0; i < n; i++)
            {
                GetRow(out a[i], out b[i]);
            }
        }

        public void GetColumn(int n, out string[] a, out string[] b, out string[] c)
        {
            a = new string[n];
            b = new string[n];
            c = new string[n];
            for (int i = 0; i < n; i++)
            {
                GetRow(out a[i], out b[i], out c[i]);
            }
        }

        public void GetColumn(int n, out string[] a, out string[] b, out string[] c, out string[] d)
        {
            a = new string[n];
            b = new string[n];
            c = new string[n];
            d = new string[n];
            for (int i = 0; i < n; i++)
            {
                GetRow(out a[i], out b[i], out c[i], out d[i]);
            }
        }

        public void GetIntColumn(int n, out int[] a)
        {
            string[] sa;
            GetColumn(n, out sa);
            a = ToIntArray(sa);
        }

        public void GetIntColumn(int n, out int[] a, out int[] b)
        {
            string[] sa, sb;
            GetColumn(n, out sa, out sb);
            a = ToIntArray(sa);
            b = ToIntArray(sb);
        }

        public void GetIntColumn(int n, out int[] a, out int[] b, out int[] c)
        {
            string[] sa, sb, sc;
            GetColumn(n, out sa, out sb, out sc);
            a = ToIntArray(sa);
            b = ToIntArray(sb);
            c = ToIntArray(sc);
        }

        public void GetIntColumn(int n, out int[] a, out int[] b, out int[] c, out int[] d)
        {
            string[] sa, sb, sc, sd;
            GetColumn(n, out sa, out sb, out sc, out sd);
            a = ToIntArray(sa);
            b = ToIntArray(sb);
            c = ToIntArray(sc);
            d = ToIntArray(sd);
        }

        public void GetLongColumn(int n, out long[] a)
        {
            string[] sa;
            GetColumn(n, out sa);
            a = ToLongArray(sa);
        }

        public void GetLongColumn(int n, out long[] a, out long[] b)
        {
            string[] sa, sb;
            GetColumn(n, out sa, out sb);
            a = ToLongArray(sa);
            b = ToLongArray(sb);
        }

        public void GetLongColumn(int n, out long[] a, out long[] b, out long[] c)
        {
            string[] sa, sb, sc;
            GetColumn(n, out sa, out sb, out sc);
            a = ToLongArray(sa);
            b = ToLongArray(sb);
            c = ToLongArray(sc);
        }

        public void GetLongColumn(int n, out long[] a, out long[] b, out long[] c, out long[] d)
        {
            string[] sa, sb, sc, sd;
            GetColumn(n, out sa, out sb, out sc, out sd);
            a = ToLongArray(sa);
            b = ToLongArray(sb);
            c = ToLongArray(sc);
            d = ToLongArray(sd);
        }

        public void GetDoubleColumn(int n, out double[] a)
        {
            string[] sa;
            GetColumn(n, out sa);
            a = ToDoubleArray(sa);
        }

        public void GetDoubleColumn(int n, out double[] a, out double[] b)
        {
            string[] sa, sb;
            GetColumn(n, out sa, out sb);
            a = ToDoubleArray(sa);
            b = ToDoubleArray(sb);
        }

        public void GetDoubleColumn(int n, out double[] a, out double[] b, out double[] c)
        {
            string[] sa, sb, sc;
            GetColumn(n, out sa, out sb, out sc);
            a = ToDoubleArray(sa);
            b = ToDoubleArray(sb);
            c = ToDoubleArray(sc);
        }

        public void GetDoubleColumn(int n, out double[] a, out double[] b, out double[] c, out double[] d)
        {
            string[] sa, sb, sc, sd;
            GetColumn(n, out sa, out sb, out sc, out sd);
            a = ToDoubleArray(sa);
            b = ToDoubleArray(sb);
            c = ToDoubleArray(sc);
            d = ToDoubleArray(sd);
        }

        #endregion

        #region get matrix

        public string[][] GetMatrix(int h)
        {
            string[][] result = new string[h][];
            for (int i = 0; i < h; i++)
            {
                result[i] = Array();
            }

            return result;

        }

        public int[][] GetIntMatrix(int h)
        {
            int[][] result = new int[h][];
            for (int i = 0; i < h; i++)
            {
                result[i] = IntArray();
            }

            return result;
        }

        public long[][] GetLongMatrix(int h)
        {
            long[][] result = new long[h][];
            for (int i = 0; i < h; i++)
            {
                result[i] = LongArray();
            }

            return result;
        }

        public double[][] GetDoubleMatrix(int h)
        {
            double[][] result = new double[h][];
            for (int i = 0; i < h; i++)
            {
                result[i] = DoubleArray();
            }

            return result;
        }

        public char[][] GetCharMatrix(int h)
        {
            char[][] result = new char[h][];
            for (int i = 0; i < h; i++)
            {
                result[i] = Next().ToCharArray();
            }

            return result;
        }

        #endregion
    }

    class Program
    {
        private int N;
        private long[] A;
        private long[] S;
        public void Solve()
        {
            var sc = new Scanner();
            N = sc.NextInt();
            A = sc.LongArray();
            S = new long[N + 1];
            for (int i = 0; i < N; i++)
            {
                S[i + 1] = S[i] + A[i];
            }

            //[0,i),[i,N)
            long ans = long.MaxValue;
            for (int i = 2; i <= N - 2; i++)
            {
                var abs = Abs(i);
                //Console.WriteLine(abs);
                ans = Math.Min(ans, abs);
                
            }
            Console.WriteLine(ans);
        }

        private int Naka(int l, int r)
        {
            int min = l;
            int max = r;
            long sum = Sum(l, r);
            while (max - min > 1)
            {
                int med = (min + max) / 2;
                if (Sum(l, med) > Sum(med, r))
                {
                    max = med;
                }
                else
                {
                    min = med;
                }
            }

            return min;
        }

        private long Abs(int i)
        {
            int aa = Naka(0, i);
            int bb = Naka(i, N);
            //Console.WriteLine($"{aa} {bb}");
            long ans = long.MaxValue;
            for (int j = -5; j < 5; j++)
            {
                int bc = aa + j;
                //Console.WriteLine("bc="+bc);
                if (bc - 0 <= 0 || i - bc <= 0) continue;
                //Console.WriteLine("bc=" + bc);
                for (int k = -5; k < 5; k++)
                {
                    int de = bb + k;
                    if(de-i<=0||N-de<=0)continue;
                    //Console.WriteLine($"{bc} {de}");
                    long b = Sum(0, bc);
                    long c = Sum(bc, i);
                    long d = Sum(i, de);
                    long e = Sum(de, N);
                    long min = Math.Min(b, Math.Min(c, Math.Min(d, e)));
                    long max = Math.Max(b, Math.Max(c, Math.Max(d, e)));
                    ans = Math.Min(ans, max - min);
                    //Console.WriteLine(ans);
                }
            }

            return ans;
        }

        /// <summary>
        /// [l,r)の和
        /// </summary>
        /// <param name="l"></param>
        /// <param name="r"></param>
        /// <returns></returns>
        private long Sum(int l, int r)
        {
            return S[r] - S[l];
        }

        static void Main(string[] args) => new Program().Solve();
    }
}