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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include<bits/stdc++.h>
using namespace std;
//#define CHKR
#define ll long long
#define pb push_back
//#define mp make_pair
#define fr first
#define sc second
#define ARRS int(2e5+500)
#define BARRS int(6e4+600)
#define MAX ((long long)(1e9+1))
#define EP ((long double)(1e-9))
#define MMAX ((long long)(1e9+10))
#define HS1 ((long long)(1000001329))
#define HS2 ((long long)(1000001531))
#define MOD ((long long)1000000009ll)
#define SQ 31622780
#define PI 3.14159265358979323846264338327950288419716939937510
#define BG 4294967232ll
#define MH 200008


#ifndef CHKR
int main(){
#else
int doit(fstream &in,fstream &out){
    cin.rdbuf(in.rdbuf());
    cout.rdbuf(out.rdbuf());
#endif
    #ifdef KHOKHO
        #ifndef CHKR
        freopen("in.in","r",stdin);
        freopen("out.out","w+",stdout);
        #endif //CHKR
    #endif //KHOKHO
    ll d=MAX,c=0;
    string s;
    ll p=0;
    string k="";
    cin>>c;
    cin>>s;
    for(int i=0; i<s.size(); i++){
        if(s[i]==')'){
            if(p)p--,k+=')';
            else k="("+k+")";
        }
        else p++,k+='(';
    }
    while(p--)k+=')';
    cout<<k;
    return 0;
}