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: Accepted)

{-# LANGUAGE BangPatterns #-}
import Control.Monad
import Control.Applicative
import Data.List
import Data.Maybe
import Data.Bits
import qualified Data.IntMap.Strict as M
import qualified Data.ByteString.Char8 as B
import qualified Data.Vector.Unboxed as V
import Data.Vector.Unboxed ((!))


findMidWithin :: V.Vector Int -> (Int,Int) -> Int
findMidWithin vec (from,to) =
    fromMaybe to $ testIndex
    <$> V.findIndex ((>= lohi) . (`shiftL` 1)) (V.slice from (to-from+1) vec)
    where
      len = V.length vec
      n = len - 1
      lohi = vec!0 + vec!n
      testIndex 0 = from
      testIndex j | lohi <= vec!(from+j-1) + vec!(from+j) = from+j-1
                  | otherwise                             = from+j

findMidIntvWithin :: V.Vector Int -> (Int,Int) -> (Int,Int) -> Int
findMidIntvWithin vec (intvFrom,intvTo) (withinFrom,withinTo)
  = intvFrom
    + findMidWithin (V.slice intvFrom (intvTo-intvFrom+1) vec)
       (withinFrom - intvFrom, withinTo - intvFrom)
                                 
query :: Int -> V.Vector Int -> Int
query n sums =
  V.minimum $ V.imap (curry $ uncurry (-) . maxmin . map diff . poss) inds
  where
    maxmin = foldl' (\(!mx,!mn) v -> (mx `max` v, mn `min` v))
                    (minBound,maxBound)
    diff (i,j) = sums!j - sums!i
    poss (j',(i,k)) = [(0,i),(i,j),(j,k),(k,n)]
      where j = j' + 2
    inds =
      V.unfoldrN (n-3)
                 (\(!i,!j,!k) ->
                    let j' = findMidIntvWithin sums (0,i) (j,i-1)
                        k' = findMidIntvWithin sums (i,n) (k,n-1)
                    in Just ((j',k'),(i+1,j',k'`max`(i+1))))
                 (2,1,3)
     
main :: IO ()
main = do
  n <- read <$> getLine
  ais <- V.unfoldrN n (B.readInt . B.dropWhile (==' ')) <$> B.getLine
  putStrLn $ show $ query n $ V.scanl (+) 0 ais