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>
// statics
using namespace std;
using int64 = long long;
using PAIR = pair<int, int>;
constexpr int INF = 1 << 30;
constexpr int64 LINF = 1LL << 60;
constexpr int MOD = 1e9 + 7;
constexpr int MAX_N = 1e5 + 1;

// init/input
#define int int64
#define INIT ios::sync_with_stdio(false);cin.tie(0);
#define VAR(type, ...) type __VA_ARGS__;MACRO_VAR_Scan(__VA_ARGS__);
template<typename T> void MACRO_VAR_Scan(T &t) {cin>>t;}
template<typename First, typename...Rest> void MACRO_VAR_Scan(First &first, Rest&...rest) {cin>>first;MACRO_VAR_Scan(rest...);}
#define VEC(type, c, n) vector<type> c(n);for(auto &&i:c)cin>>i;

// out
#define OUT(dist) cout<<(dist);
#define FOUT(n, dist) cout<<fixed<<setprecision(n)<<(dist);
#define SP cout<<" ";
#define BR cout<<"\n";
#define debug(x) cerr << #x << ":" << (x);BR;

// utility
#define ALL(a) (a).begin(), (a).end()
#define EACH(i, a) for(auto &&i:(a))
#define FOR(i, a, b) for(int i=(a);i<(b);++i)
#define RFOR(i, a, b) for(int i=(b)-1;i>=0;--i)
#define REP(i, n) for(int i=0;i<(n);++i)
#define RREP(i, n) for(int i=(n)-1;i>=0;--i)

signed main() {
  INIT;

	VAR(int, n);
	VAR(string, s);

	int l = 0, r = 0, cnt = 0;
	EACH(si, s) {
		if (si == '(') {
			cnt++;
		} else {
			cnt--;
			if (cnt < 0) {
				l++;
				cnt = 0;
			}
			
		}
	}
	if (cnt > 0) r = cnt;
	if (cnt < 0) l += cnt;

	OUT(string(l, '('));
	OUT(s);
	OUT(string(r, ')'))BR;
  
  return 0;
}