Problem:
Score : 400 points

Problem Statement

You are given a string S of length N consisting of ( and ). Your task is to insert some number of ( and ) into S to obtain a correct bracket sequence.

Here, a correct bracket sequence is defined as follows:

() is a correct bracket sequence.

If X is a correct bracket sequence, the concatenation of (, X and ) in this order is also a correct bracket sequence.

If X and Y are correct bracket sequences, the concatenation of X and Y in this order is also a correct bracket sequence.

Every correct bracket sequence can be derived from the rules above.

Find the shortest correct bracket sequence that can be obtained. If there is more than one such sequence, find the lexicographically smallest one.

Constraints

The length of S is N.

1 ≤ N ≤ 100

S consists of ( and ).

Input

Input is given from Standard Input in the following format:

N
S

Output

Print the lexicographically smallest string among the shortest correct bracket sequences that can be obtained by inserting some number of ( and ) into S.

Sample Input 1

3
())

Sample Output 1

(())

Sample Input 2

6
)))())

Sample Output 2

(((()))())

Sample Input 3

8
))))((((

Sample Output 3

(((())))(((())))

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

import Data.List

main = do
 getLine
 as <- getLine
 putStrLn (ins as)

ins as
 | valid as == (Just 0) = as
 | otherwise = replicate nbl '(' ++ as ++ replicate nbr ')'
  where
   nbl = let (bl,br) = partition (=='(') (dropWhile (==')') as)
          in length (takeWhile (==')') as) + max 0 (length br - length bl)
   nbr = let (bl,br) = partition (=='(') as in (length bl) + nbl - (length br)
--   nbr = let (br,bl) = partition (==')') $ dropWhile (=='(') (reverse as)
--          in length (takeWhile (=='(') (reverse as)) + length bl - length br

valid = foldr (\b acc -> if b==')' then succ <$> acc else if b=='(' && ((Just 0) < acc) then pred <$> acc else Nothing) (Just 0)