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 <iostream>
#include <sstream>
#include <algorithm>
#include <string>
#include <vector>
#include <map>
#include <set>
#include <queue>
#include <deque>
#include <stack>
#include <memory>
#include <complex>
#include <numeric>
#include <cstdio>
#include <iomanip>
#include <random>

#define REP(i,m,n) for(int i=int(m);i<int(n);i++)
#define RREP(i,m,n) for(int i=int(n)-1;i>=int(m);--i)
#define EACH(i,c) for (auto &(i): c)
#define all(c) begin(c),end(c)
#define EXIST(s,e) ((s).find(e)!=(s).end())
#define SORT(c) sort(begin(c),end(c))
#define pb emplace_back
#define MP make_pair
#define SZ(a) int((a).size())

#ifdef LOCAL
#define DEBUG(s) cout << (s) << endl
#define dump(x)  cerr << #x << " = " << (x) << endl
#define BR cout << endl;
#else
#define DEBUG(s) do{}while(0)
#define dump(x) do{}while(0)
#define BR 
#endif
using namespace std;

using UI = unsigned int;
using UL = unsigned long;
using LL = long long int;
using ULL = unsigned long long;
using VI = vector<int>;
using VVI = vector<VI>;
using VLL = vector<LL>;
using VVLL = vector<VLL>;
using VS = vector<string>;
using PII = pair<int,int>;
using VP = vector<PII>;

void solve() {
    int n;
    cin >> n;
    VLL a(n);
    REP(i,0,n) cin >> a[i];
    VLL ru(n+1);
    REP(i,0,n) ru[i+1] = ru[i] + a[i];
    LL ans = 1LL << 60;
    REP(i,1,n-2) {
        LL lsum = ru[i+1], rsum = ru[n] - ru[i+1];
        auto lit = lower_bound(all(ru), lsum/2);
        auto rit = lower_bound(all(ru), lsum + rsum / 2);
        int li = distance(ru.begin(), lit);
        int ri = distance(ru.begin(), rit);
        VVLL c(4);
        REP(j,0,2) {
            c[j*2].push_back(ru[li]);
            c[j*2].push_back(ru[i+1] - ru[li]);
            c[j].push_back(ru[ri] - ru[i+1]);
            c[j].push_back(ru[n] - ru[ri]);

            c[j*2+1].push_back(ru[li-1]);
            c[j*2+1].push_back(ru[i+1] - ru[li-1]);
            c[j+2].push_back(ru[ri-1] - ru[i+1]);
            c[j+2].push_back(ru[n] - ru[ri-1]);
        }
        REP(j,0,4) {
            LL diff = *max_element(all(c[j])) - *min_element(all(c[j]));
            ans = min(ans, diff);
        }
    }
    cout << ans << endl;
}

int main() {
    solve();
    
    return 0;
}