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
# （しゃくとりだと各境界を1ずらすたびに区間和を増減できるので必須ではないが、
#  計算しておいたほうがコードが簡単）
accum = as.dup
(N-1).times do |i|
    accum[i+1] += accum[i]
end

ans = 10**20

# [0, div1], [div1+1, div2], [div2+1, div3], [div3+1, N-1] に分ける
div1, div3 = 0, 2
(1..(N-3)).each do |div2|

    # [0, div1]と[div1+1, div2]が半分に分かれるようにdiv1移動（しゃくとり）
    while div1+1 < div2
        diff1 = (accum[div2] - 2*accum[div1]).abs
        diff2 = (accum[div2] - 2*accum[div1+1]).abs
        if diff1 > diff2
            div1 += 1
        else
            break
        end
    end

    # [div2+1, div3]と[div3+1, N-1]が半分に分かれるようにdiv3移動（しゃくとり）
    while div3+1 < N-1
        diff1 = (accum[N-1] - 2*accum[div3] + accum[div2]).abs
        diff2 = (accum[N-1] - 2*accum[div3+1] + accum[div2]).abs
        if diff1 > diff2
            div3 += 1
        else
            break
        end
    end

    parts = [
        accum[N-1]  - accum[div3],
        accum[div3] - accum[div2],
        accum[div2] - accum[div1],
        accum[div1]
    ]

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