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 Lua (5.3.2) (judge verdict: Accepted)

local mfl, mma, mmi = math.floor, math.max, math.min
local function lower_bound(ary, x, min, max)
  if(x <= ary[min]) then return min end
  if(ary[max] < x) then return max + 1 end
  while(1 < max - min) do
    local mid = mfl((min + max) / 2)
    if(ary[mid] < x) then
      min = mid
    else
      max = mid
    end
  end
  return max
end

local ret = nil
local n = io.read("*n")
local tot = {}
local cur = 0
for i = 1, n do
  cur = cur + io.read("*n")
  tot[i] = cur
end
for i = 3, n - 1 do
  local leftsum = tot[i - 1]
  local rightsum = tot[n] - leftsum
  local leftlb = lower_bound(tot, leftsum / 2, 1, i - 1)
  leftlb = mmi(leftlb, i - 2)
  local rightlb = lower_bound(tot, leftsum + rightsum / 2, i, n)
  rightlb = mmi(rightlb, n - 1)
  for k = 1, 4 do
    local v1 = tot[leftlb]
    local v3 = tot[rightlb] - leftsum
    if k <= 2 then
      if(1 < leftlb) then
        v1 = tot[leftlb - 1]
      end
    end
    if k % 2 == 0 then
      if(i < rightlb) then
        v3 = tot[rightlb - 1] - leftsum
      end
    end
    local v2 = leftsum - v1
    local v4 = rightsum - v3
    local mi = mmi(v1, mmi(v2, mmi(v3, v4)))
    local ma = mma(v1, mma(v2, mma(v3, v4)))
    if(ret == nil) then
      ret = ma - mi
    else
      ret = mmi(ret, ma - mi)
    end
  end
end
print(ret)