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 int long long
// #define double long double
#define FOR(i, a, b) for(ll i = (a); i < (b); ++i)
#define FORR(i, a, b) for(ll i = (a); i > (b); --i)
#define REP(i, n) for(ll i = 0; i < (n); ++i)
#define REPR(i, n) for(ll i = n; i >= 0; i--)
#define FOREACH(x, a) for(auto &(x) : (a))
#define VECCIN(x)                                                              \
    for(auto &youso_ : (x)) cin >> youso_
#define bitcnt __builtin_popcount
#define SZ(x) ((ll)(x).size())
#define fi first
#define se second
#define All(a) (a).begin(), (a).end()
#define rAll(a) (a).rbegin(), (a).rend()
template <typename T = long long> inline T IN() {
    T x;
    cin >> x;
    return (x);
}
inline void CIN() {}
template <class Head, class... Tail>
inline void CIN(Head &&head, Tail &&... tail) {
    cin >> head;
    CIN(move(tail)...);
}
#define CCIN(...)                                                              \
    char __VA_ARGS__;                                                          \
    CIN(__VA_ARGS__)
#define DCIN(...)                                                              \
    double __VA_ARGS__;                                                        \
    CIN(__VA_ARGS__)
#define LCIN(...)                                                              \
    ll __VA_ARGS__;                                                            \
    CIN(__VA_ARGS__)
#define SCIN(...)                                                              \
    string __VA_ARGS__;                                                        \
    CIN(__VA_ARGS__)
#define Printv(v)                                                              \
    {                                                                          \
        REP(x, v.size()) { cout << v[x] << (x == v.size() - 1 ? "\n" : " "); } \
    }
template <typename T = string> inline void eputs(T s) {
    cout << s << "\n";
    exit(0);
}
template <typename A, size_t N, typename T>
void Fill(A (&array)[N], const T &val) {
    std::fill((T *)array, (T *)(array + N), val);
}
template <typename T> using PQG = priority_queue<T, vector<T>, greater<T>>;
template <typename T> using PQ = priority_queue<T>;

typedef long long ll;
typedef pair<ll, ll> PL;
typedef vector<PL> VPL;
typedef vector<ll> VL;
typedef vector<VL> VVL;
typedef vector<double> VD;

const int INF = 1e9;
const int MOD = 1e9 + 7;
const ll LINF = 1e18;
const ll dx[] = {1, 0, -1, 0};
const ll dy[] = {0, 1, 0, -1};
#define PI 3.141592653589793238

ll used[100001];

/*
    string中の特定文字列をstringで置換する
*/
string ReplaceString(string String1 // 置き換え対象
                     ,
                     string String2 // 検索対象
                     ,
                     string String3 // 置き換える内容
) {
    string::size_type Pos(String1.find(String2));
    bool flag = false;
    while(Pos != string::npos) {
        String1.replace(Pos, String2.length(), String3);
        Pos = String1.find(String2, Pos + String3.length());
    }

    return String1;
}

signed main() {
    LCIN(N);
    SCIN(S);
    ll cnt1 = 0, cnt2 = 0, f = 0, b = 0;
    REPR(i, N - 1) {
        if(S[i] == '(') {
            if(b)
                b--;
            else
                cnt1++;
        } else {
            b++;
        }
    }
    REP(i, N) {
        if(S[i] == ')') {
            if(f)
                f--;
            else
                cnt2++;
        } else {
            f++;
        }
    }
    S = string(cnt2, '(') + S + string(cnt1, ')');
    cout << S << "\n";
}