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: Wrong Answer)

#![allow(unused_imports, unused_macros, dead_code)]
use std::f64::*;
use std::cmp::*;
use std::collections::*;
use std::ops::*;

macro_rules! dump{
    ($($a:expr),*) => {
        #[cfg(debug_assertions)]
        eprintln!(
            concat!("{}:{}:{}: ",$(stringify!($a), " = {:?}, "),*),
            file!(), line!(), column!(), $($a),*
        );
        #[cfg(not(debug_assertions))]
        {};
    }
}

const INF: i64 = 0x3f3f3f3f3f3f3f3f;
const MOD: i64 = 1000000007;
const NOTHING: usize = std::usize::MAX;
const DX: [usize; 4] = [1, 0, std::usize::MAX, 0];
const DY: [usize; 4] = [0, 1, 0, std::usize::MAX];

fn solve<R: std::io::Read, W: std::io::Write>(io: &mut IO<R, W>) {
    let n: usize = io.read();
    let xs: Vec<usize> = io.reads(n);
    let mut bit = BIT::new(n);
    for i in 0..n {
        bit.add(i, xs[i] as i64);
    }
    let s = bit.sum(0..n) / 4;
    let x = lower_bound(0, n, |x| bit.sum(0..x) >= s);
    let y = lower_bound(0, n, |x| bit.sum(0..x) >= 2 * s);
    let z = lower_bound(0, n, |x| bit.sum(0..x) >= 3 * s);

    let d: usize = 150;
    let lx = max(0, x as isize - d as isize) as usize;
    let rx = min(n, x + d);

    let ly = max(0, y as isize - d as isize) as usize;
    let ry = min(n, y + d);

    let lz = max(0, z as isize - d as isize) as usize;
    let rz = min(n, z + d);

    let mut res = std::i64::MAX;
    for i in lx..rx {
        for j in max(i + 1, ly)..min(n, max(i + 1, ly) + 2 * d) {
            for k in max(j + 1, lz)..min(n, max(j + 1, lz) + 2 * d) {
                let p = bit.sum(0..i);
                let q = bit.sum(i..j);
                let r = bit.sum(j..k);
                let s = bit.sum(k..n);
                let m = min(min(p, q), min(r, s));
                let mm = max(max(p, q), max(r, s));
                res = min(res, mm - m);
            }
        }
    }
    io.writeln(res);
}

pub fn lower_bound<F: FnMut(usize) -> bool>(low: usize, high: usize, mut p: F) -> usize {
    debug_assert!(p(high));
    let mut l: usize = low;
    let mut h: usize = high;
    while l < h {
        let m = l + ((h - l) >> 1);
        if p(m) {
            h = m;
        } else {
            l = m + 1;
        }
    }
    h
}

pub struct BIT {
    size: usize,
    data: Vec<i64>,
}

impl BIT {
    pub fn new(n: usize) -> BIT {
        BIT {
            size: n + 1,
            data: vec![0;n+1],
        }
    }

    // sum [0, k)
    pub fn sum_to(&self, k: usize) -> i64 {
        let mut res = 0;
        let mut i = k;
        while i > 0 {
            res += self.data[i];
            i -= i & 0usize.wrapping_sub(i);
        }
        res
    }

    // sum [l, r)
    pub fn sum(&self, range: std::ops::Range<usize>) -> i64 {
        self.sum_to(range.end) - self.sum_to(range.start)
    }

    pub fn add(&mut self, k: usize, x: i64) {
        let mut i = k + 1;
        while i < self.size {
            self.data[i] += x;
            i += i & 0usize.wrapping_sub(i);
        }
    }
}
fn main() {
    use std::io::*;
    std::thread::Builder::new()
        .stack_size(32 * 1024 * 1024)
        .spawn(|| {
            let stdin = stdin();
            let stdout = stdout();
            let mut io = IO::new(BufReader::new(stdin.lock()), BufWriter::new(stdout.lock()));
            solve(&mut io);
            io.flush();
        })
        .unwrap()
        .join()
        .unwrap();
}

pub struct IO<R, W> {
    reader: std::io::Bytes<R>,
    writer: W,
}

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> IO<R, W> {
        IO {
            reader: r.bytes(),
            writer: w,
        }
    }

    pub fn read<T: Read>(&mut self) -> T {
        T::read(&mut self.reader)
    }

    pub fn reads<T: Read>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| T::read(&mut self.reader)).collect()
    }

    pub fn write<T: std::fmt::Display>(&mut self, x: T) {
        self.writer.write_fmt(format_args!("{}", x)).unwrap();
    }
    pub fn writeln<T: std::fmt::Display>(&mut self, x: T) {
        self.writer.write_fmt(format_args!("{}\n", x)).unwrap();
    }

    pub fn writes<T: std::fmt::Display>(&mut self, xs: &Vec<T>, sep: &str) {
        let mut it = xs.iter();
        for &x in it.next().iter() {
            self.write(x);
        }
        for x in it {
            self.writer.write(sep.as_bytes()).unwrap();
            self.write(x);
        }
    }

    pub fn writesln<T: std::fmt::Display>(&mut self, xs: &Vec<T>, sep: &str) {
        self.writes(xs, sep);
        self.writer.write(b"\n").unwrap();
    }

    pub fn flush(&mut self) {
        self.writer.flush().unwrap()
    }
}

pub trait Read {
    fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self;
}

#[inline(always)]
pub fn is_whitespace(x: u8) -> bool {
    x < 33u8 || 126u8 < x
}

macro_rules! read_int_impl{
    ($($t:ty)*) => ($(
        impl Read for $t {
            fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
                let head: u8 = bytes
                    .map(|x| x.unwrap())
                    .skip_while(|&x| is_whitespace(x))
                    .next()
                    .unwrap();
                if head == 45u8 {
                    bytes.map(|x| x.unwrap())
                        .take_while(|&x| !is_whitespace(x))
                        .fold(0, |a, x| 10 * a - x as $t + 48)
                } else {
                    bytes.map(|x| x.unwrap())
                        .take_while(|&x| !is_whitespace(x))
                        .fold((head - 48) as $t, |a, x| 10 * a + x as $t - 48)
                }
            }
        }
    )*)
}
read_int_impl! { isize i8 i16 i32 i64 usize u8 u16 u32 u64}

macro_rules! read_float_impl{
    ($($t:ty)*) => ($(
        impl Read for $t {
            fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
                String::read(bytes).as_str().parse::<$t>().unwrap()
            }
        }
    )*)
}
read_float_impl! { f32 f64 }

impl Read for char {
    fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
        bytes.map(|x| x.unwrap())
            .skip_while(|&x| is_whitespace(x))
            .next()
            .unwrap() as char
    }
}

impl Read for String {
    fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
        bytes.map(|x| x.unwrap())
            .skip_while(|&x| is_whitespace(x))
            .take_while(|&x| !is_whitespace(x))
            .map(|x| x as char)
            .collect()
    }
}

impl Read for Vec<char> {
    fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
        bytes.map(|x| x.unwrap())
            .skip_while(|&x| is_whitespace(x))
            .take_while(|&x| !is_whitespace(x))
            .map(|x| x as char)
            .collect()
    }
}

macro_rules! read_tuple_impl{
    ($($t:ident),+) => {
        impl<$($t: Read),*> Read for ($($t),*) {
            fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
                ($($t::read(bytes)),+)
            }
        }
    }
}
read_tuple_impl!(A, B);
read_tuple_impl!(A, B, C);
read_tuple_impl!(A, B, C, D);
read_tuple_impl!(A, B, C, D, E);