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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace AtCoder
{
    class Program
    {
        private const int MODULO = 1000000007;
        private static string Read() { return Console.ReadLine(); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }

        static IEnumerable<int> Count(IEnumerable<char> S)
        {
            int N = S.Count();
            var counts = new int[N];
            for (int i = 0; i < N; ++i)
            {
                int prev = (i == 0) ? 0 : counts[i - 1];
                switch (S.ElementAt(i))
                {
                case '(': counts[i] = prev + 1; break;
                case ')': counts[i] = prev - 1; break;
                }
            }
            return counts;
        }
        static string Solve(string S)
        {
            int min = Math.Min(Count(S).Min(), 0);
            int max = Math.Max(Count(S.Reverse()).Max(), 0);
            return new string('(', -min) + S + new string(')', max);
        }
        static void Main(string[] args)
        {
            int N = ReadInt();
            string S = Read();
            Console.WriteLine(Solve(S));
        }
    }
}