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: Time Limit Exceeded)

import Control.Arrow
import Data.List
import qualified Data.ByteString.Char8 as B
import Data.Ord
import Data.Function
import Debug.Trace

main :: IO ()
main = getLine >> breads >>= print . solve 

solve :: [Int] -> Int
solve as = loop (s, []) $ zip as ss 
    where
        s = sum as
        ss = scanl1 (+) as

        loop (acc, ps) (p@(a, t) : qs) = loop (min acc val, p : ps) qs
            where
                (b, c) = split t (p : ps)
                (d, e) = split (s - t) qs
                vs = sort [b, c, d, e]
                val = last vs - head vs
        loop (acc, _) _ = acc

        split t = check 0
            where
                check acc ((a, _) : xs)
                    | ((>=) `on` absf) acc next = check next xs
                    | otherwise = (acc, t - acc)
                    where
                        next = acc + a
                check acc _ = (acc, 0)
                absf = abs . (t -) . (2 *)


breads :: IO [Int]
breads = unfoldr (\s -> second (maybe B.empty snd . B.uncons) <$> B.readInt s) <$> B.getLine