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: Wrong Answer)

n = gets.chomp.to_i
a = gets.chomp.split("\s").map(&:to_i)

# p a

c1 = 0
c2 = n - 1

result = Array.new

while true do
    tmp_result = Array.new
    
    # puts "c1: #{c1}, c2: #{c2}"

    v1 = a[0..c1].inject(:+)
    v2 = a[c2..n-1].inject(:+)

    for i in c1+1...c2-1 do
        # puts "i: #{i}"
        v3 = a[c1+1..i].inject(:+)
        v4 = a[i+1..c2-1].inject(:+)
        tmp_result.push [v1, v2, v3, v4].max - [v1, v2, v3, v4].min
        # puts "v1: #{v1}, v2: #{v2}, v3: #{v3}, v4: #{v4}"
        # puts "differ: " + ([v1, v2, v3, v4].max - [v1, v2, v3, v4].min).to_s
    end

    if result.length == 0 then
        result.push tmp_result.min
    elsif result.min > tmp_result.min then
        result.push tmp_result.min
    else
        break
    end

    if c1+2 == c2-1 then
        break
    end

    next_v1 = v1 + a[c1]
    next_v2 = v2 + a[c2-1]

    if next_v1 > next_v2 then
        c2 -= 1
    else
        c1 += 1
    end

end

puts result.min