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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <algorithm>
#include <cctype>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <iostream>
#include <map>
#include <queue>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <utility>
#include <vector>
using namespace std;

typedef long long s64;
s64 sumArray[200000];
int N;

void calcLR(int first, int last, int leftLast, s64* left, s64* right)
{
    *left = sumArray[leftLast];
    if (first != 0)
    {
        *left -= sumArray[first - 1];
    }
    *right = sumArray[last] - sumArray[leftLast];
}

int findLeftLast(int first, int last, int candFirst, int candLast, int depth)
{
    if (candFirst == candLast)
    {
        return candFirst;
    }
    if (candLast - candFirst == 1)
    {
        s64 left, right;
        calcLR(first, last, candFirst, &left, &right);
        s64 diff1 = abs(left - right);
        calcLR(first, last, candLast, &left, &right);
        s64 diff2 = abs(left - right);
        if (diff1 < diff2)
        {
            return candFirst;
        }
        else
        {
            return candLast;
        }
    }
    int llast = (candFirst + candLast) / 2;
    s64 left, right;
    calcLR(first, last, llast, &left, &right);
    if (left == right)
    {
        return llast;
    }
    else if (left < right)
    {
        if (llast == last - 1)
        {
            return llast;
        }
        return findLeftLast(first, last, llast, candLast, depth + 1);
    }
    else
    {
        if (llast == 0)
        {
            return llast;
        }
        return findLeftLast(first, last, candFirst, llast, depth + 1);
    }
}

int main()
{
    cin >> N;
    for (int i = 0; i < N; ++i)
    {
        int a;
        cin >> a;
        if (i == 0)
        {
            sumArray[i] = a;
        }
        else
        {
            sumArray[i] = sumArray[i - 1] + a;
        }
    }

    auto calc = [](int first, int last, s64* min, s64* max) {
        int leftLast = findLeftLast(first, last, first, last - 1, 0);
        s64 left, right, diff;
        calcLR(first, last, leftLast, &left, &right);
        if (left < right)
        {
            *min = left;
            *max = right;
        }
        else
        {
            *min = right;
            *max = left;
        }
    };

    s64 diffMin = -1;
    for (int leftLast = 1; leftLast <= N - 3; ++leftLast)
    {
        s64 leftMin, leftMax, rightMin, rightMax;
        calc(0, leftLast, &leftMin, &leftMax);
        calc(leftLast + 1, N - 1, &rightMin, &rightMax);
        s64 diff = max(leftMax, rightMax) - min(leftMin, rightMin);
        if (diffMin == -1 || diff < diffMin)
        {
            diffMin = diff;
        }
    }
    cout << diffMin << endl;

    return 0;
}