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 Ruby (2.3.3) (judge verdict: Accepted)

N = gets.to_i
as = gets.split.map(&:to_i)

# 区間の和を使うので累積和を前計算
accum = as.dup
(N-1).times do |i|
    accum[i+1] += accum[i]
end

ans = 10**20

# 区間2つ+区間2つの境界を全探索
# [0, center] と [center+1, N-1] に分ける
(1..(N-3)).each do |center|
    sum_l = accum[center]
    sum_r = accum[N-1] - accum[center]
    parts = []

    # s1を半分に分ける。
    mid = (1..center).bsearch{|i| accum[i]*2 > sum_l}
    s11, s21 = accum[mid], accum[mid-1]
    s12, s22 = sum_l - s11, sum_l - s21
    if (s11 - s12).abs < (s21 - s22).abs
        parts.push(s11, s12)
    else
        parts.push(s21, s22)
    end

    # s2を半分に分ける
    mid = ((center+2)...N).bsearch{|i| (accum[i] - sum_l)*2 > sum_r}
    s11, s21 = accum[mid]-sum_l, accum[mid-1]-sum_l
    s12, s22 = sum_r - s11, sum_r - s21
    if (s11 - s12).abs < (s21 - s22).abs
        parts.push(s11, s12)
    else
        parts.push(s21, s22)
    end

    result = parts.max - parts.min
    ans = result if ans > result
end
puts ans