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: Wrong Answer)

#include <iostream>
#include <string>
#include <vector>
// #include <unordered_map>
#include <algorithm>
#include <sstream>
#include <queue>
#include <deque>
#include <bitset>
#include <iterator>
#include <list>
#include <stack>
#include <map>
#include <set>
#include <functional>
#include <numeric>
#include <utility>
#include <limits>
#include <time.h>
#include <math.h>
#include <stdio.h>
#include <string.h>
#include <stdlib.h>
#include <assert.h>
#include <climits>

using namespace std;

#define SCD(t) scanf("%d",&t)
#define SCLD(t) scanf("%ld",&t)
#define SCLLD(t) scanf("%lld",&t)
#define SCC(t) scanf("%c",&t)
#define SCS(t) scanf("%s",t)
#define SCF(t) scanf("%f",&t)
#define SCLF(t) scanf("%lf",&t)
#define FOR(i, j, k) for(int i=j ; i<k ; i++)
#define RFOR(i, j, k) for(int i=j ; i>=k ; i--)
#define REP(i, j) FOR(i, 0, j)
#define RREP(i, j) RFOR(i, j, 0)
#define PI 3.1415926535897932384626433832795
typedef long long int ll;
typedef unsigned long long int  ull;
typedef pair<int, int> pii;
typedef vector<int> vi;
typedef vector<string> vs;
typedef vector<pii> vii;
typedef vector<ll> vll;
typedef vector<vi> vvi;

int main() {

    int N; cin >> N;
    string s; cin >> s;
    vi a(N);

    int l = 0, r = 0, p = 0;
    int lm = 0;
    REP(i, N) {

        if (s[i] == '(') l++;
        else r++;
        a[i] = l - r;
        lm = min(lm, a[i]);

    }

    lm -= a[N-1];
    lm = min(lm, 0);

    REP(i, r-l-lm) cout << '(';
    cout << s;
    REP(i, l-r-lm) cout << ')';
    cout << endl;
    
    return 0;
}