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 Haskell (GHC 7.10.3) (judge verdict: Wrong Answer)


equalCut ls@(l:lls) rs@(r:lrs) lsum rsum lacc racc n len
  | n == len = (reverse lacc, racc)
  | lsum < rsum =  equalCut lls rs (l+lsum) rsum (l:lacc) racc n (len+1)
  | lsum >= rsum = equalCut ls lrs lsum (r+rsum) lacc (r:racc) n (len+1)
  | otherwise = undefined

getEqCuts0 as = equalCut las lrs (a0+a1) (r0+r0) [a1,a0] [r1,r0] (length as-4) 0
  where rs = reverse as
        (a0:a1:las) = as
        (r0:r1:lrs) = rs

getEqCuts1 as = equalCut las lrs a r [a] [r] (length as-2) 0
  where rs = reverse as
        (a:las) = as
        (r:lrs) = rs

solve0 as = [as0, as1, as2, as3]
  where (ls, rs) = getEqCuts0 as
        (as0, as1) = getEqCuts1 ls
        (as2, as3) = getEqCuts1 rs

solveA [as0, as1, as2, as3] = [as0_, as1__, as2__, as3_]
  where (as1_, as2_) = getEqCuts1 (as1++as2)
        (as0_, as1__) = getEqCuts1 (as0++as1_)
        (as2__, as3_) = getEqCuts1 (as2_++as3)

solve ass
  | ass == nss = ass
  | otherwise = solve nss
  where nss = solveA ass

main :: IO ()
main = do
  _ <- read <$> getLine :: IO Int
  as <- map read.words <$> getLine :: IO [Int]
  let ss = map sum $ solve (solve0 as)
  print $ maximum ss - minimum ss