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 Rust (1.15.1) (judge verdict: Accepted)

//  ____            _                  _                       _       _
// |  _ \ _   _ ___| |_    ___ _ __   | |_ ___ _ __ ___  _ __ | | __ _| |_ ___
// | |_) | | | / __| __|  / __| '_ \  | __/ _ \ '_ ` _ \| '_ \| |/ _` | __/ _ \
// |  _ <| |_| \__ \ |_  | (__| |_) | | ||  __/ | | | | | |_) | | (_| | ||  __/
// |_| \_\\__,_|___/\__|  \___| .__/___\__\___|_| |_| |_| .__/|_|\__,_|\__\___|
//                            |_| |_____|               |_|

//https://github.com/manta1130/Competitive_Programming_Template_Rust

#[macro_use]
mod input {

    use std;
    use std::io;

    const SPLIT_DELIMITER: char = ' ';

    #[macro_export]
    #[allow(unused_macros)]
    macro_rules! input {
    ( $($x:expr ),*) => {
        {
            let temp_str = input_line_str();
            let mut split_result_iter = temp_str.split_whitespace();
                $(
                let buf_split_result = split_result_iter.next();
                let buf_split_result = buf_split_result.unwrap();
                    ($x) = buf_split_result.parse().unwrap();
                )*
        }
    };
}

    #[allow(dead_code)]
    pub fn input_line_str() -> String {
        let mut s = String::new();
        io::stdin().read_line(&mut s).unwrap();
        s.trim().to_string()
    }

    #[allow(dead_code)]
    pub fn p<T>(t: T)
    where
        T: std::fmt::Display,
    {
        println!("{}", t);
    }

    #[allow(dead_code)]
    pub fn input_vector2d<T>(line: usize) -> Vec<Vec<T>>
    where
        T: std::str::FromStr,
    {
        let mut v: Vec<Vec<T>> = Vec::new();

        for _ in 0..line {
            let vec_line = input_vector();
            v.push(vec_line);
        }
        v
    }

    #[allow(dead_code)]
    pub fn input_vector<T>() -> Vec<T>
    where
        T: std::str::FromStr,
    {
        let mut v: Vec<T> = Vec::new();

        let s = input_line_str();
        let split_result = s.split(SPLIT_DELIMITER);
        for z in split_result {
            let buf = match z.parse() {
                Ok(r) => r,
                Err(_) => panic!("Parse Error"),
            };
            v.push(buf);
        }
        v
    }

    #[allow(dead_code)]
    pub fn input_vector_row<T>(n: usize) -> Vec<T>
    where
        T: std::str::FromStr,
    {
        let mut v = Vec::with_capacity(n);
        for _ in 0..n {
            let buf = match input_line_str().parse() {
                Ok(r) => r,
                Err(_) => panic!("Parse Error"),
            };
            v.push(buf);
        }
        v
    }

    pub trait ToCharVec {
        fn to_charvec(&self) -> Vec<char>;
    }

    impl ToCharVec for String {
        fn to_charvec(&self) -> Vec<char> {
            self.to_string().chars().collect::<Vec<_>>()
        }
    }
}

mod vectools {

///C++のupper_boundと同等の実装
pub fn upper_bound<T>(v: &Vec<T>, k: T) -> usize
where
    T: PartialOrd,
{
    let mut from = 0;
    let mut to = v.len();

    while to - from > 1 {
        let mid = (from + to) / 2;
        if v[mid] <= k {
            from = mid;
        } else {
            to = mid;
        }
    }
    if v[from] > k {
        from
    } else {
        to
    }
}

///C++のlower_boundと同等の実装
pub fn lower_bound<T>(v: &Vec<T>, k: T) -> usize
where
    T: PartialOrd,
{
    let mut from = 0;
    let mut to = v.len();

    while to - from > 1 {
        let mid = (from + to) / 2;
        if v[mid] < k {
            from = mid;
        } else {
            to = mid;
        }
    }

    if v[from] < k {
        to
    } else {
        from
    }
}

///次の順列を求める。
pub fn next_permutation<T>(v: &mut Vec<T>) -> bool
where
    T: PartialOrd,
{
    let mut pivot = v.len() - 1;
    loop {
        if pivot == 0 {
            return false;
        }
        if v[pivot] > v[pivot - 1] {
            pivot -= 1;
            break;
        }
        pivot -= 1;
    }
    let mut pivot_swap = v.len() - 1;
    while v[pivot] >= v[pivot_swap] {
        pivot_swap -= 1;
    }
    v.swap(pivot, pivot_swap);
    v[pivot + 1..].reverse();
    true
}

}

use input::*;
use vectools::*;

fn main() {
    let n: usize;
    input!(n);
    let a = input_vector::<isize>();
    let mut sum = 0;
    let mut s = vec![0];
    for i in a {
        sum += i;
        s.push(sum);
    }

    let mut ans = isize::max_value();

    for j in 2..s.len() - 1 {
        let mut i = upper_bound(&s, s[j] / 2);

        for _ in 0..5 {
            if s[i] != s[j] / 2 && i != 0 {
                if (s[i] - s[j] + s[i]).abs() > (s[i - 1] - s[j] + s[i - 1]).abs() {
                    i -= 1;
                }
            }
        }

        if i == j {
            i -= 1;
        }
        if i == 0 {
            i += 1;
        }
        let mut k = upper_bound(&s, (s[j] + *s.last().unwrap()) / 2);

        for _ in 0..5 {
            if s[j] != *s.last().unwrap() / 2 && k != s.len() - 1 && j != k {
                if (s[k] - s[j] - *s.last().unwrap() + s[k]).abs()
                    > (s[k - 1] - s[j] - *s.last().unwrap() + s[k - 1]).abs()
                {
                    k -= 1;
                }
            }
        }

        if k == j {
            k += 1;
        }
        if k == s.len() - 1 {
            k -= 1;
        }
        let mut buf = vec![];
        buf.push(s[i]);
        buf.push(s[j] - s[i]);
        buf.push(s[k] - s[j]);
        buf.push(*s.last().unwrap() - s[k]);
        ans = std::cmp::min(ans, *buf.iter().max().unwrap() - *buf.iter().min().unwrap());
    }
    println!("{}", ans);
}