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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.*;
import java.util.*;
import java.util.function.BiFunction;
import java.util.function.Consumer;
import java.util.function.Supplier;

public class Main {
    public static void main(String[] args) {
        final FastScanner fs = new FastScanner();
        PrintWriter out = new PrintWriter(System.out);

        int N = fs.nextInt();
        String S = fs.next();
        int clauseStartCount = 0;
        int clauseEndCount = 0;
        Stack<Character> stack = new Stack();
        for (int i = 0; i < S.length(); i++) {
            char c = S.charAt(i);
            if (c == '(') {
                stack.push(c);
            } else {
                if (stack.isEmpty()) {
                    clauseStartCount++;
                } else {
                    char top = stack.pop();
                    if (top != '(') {
                        clauseStartCount++;
                    }
                }
            }
        }
        clauseEndCount += stack.size();

        StringBuilder answer = new StringBuilder();
        for (int i = 0; i < clauseStartCount; i++) {
            answer.append('(') ;
        }
        answer.append(S);
        for (int i = 0; i < clauseEndCount; i++) {
            answer.append(')') ;
        }


        out.println(answer.toString());
        out.flush();
    }

    static void debug(Object... os) {
        System.err.println(Arrays.deepToString(os));
    }
    // Execute func for all patterns can be represented with n bit.
    // O(n2^n)
    static void forAllBit(int n, Consumer<List<Boolean>> func) {
        for (int i = 0; i < (1 << n); i++) {
            List<Boolean> arg = new ArrayList();
            for (int j = 0; j < n; j++) {
                boolean isJthBitSet = ((i >> j) & 1) == 1;
                arg.add(isJthBitSet);
            }
            func.accept(arg);
        }
    }

    static class Pair<F extends Comparable<F>, S extends Comparable<S>> implements Comparable<Pair<F, S>> {
        F f;
        S s;

        Pair() {
        }

        Pair(F f, S s) {
            this.f = f;
            this.s = s;
        }

        Pair(Pair<F, S> p) {
            f = p.f;
            s = p.s;
        }

        @Override
        public int compareTo(Pair<F, S> p) {
            if (f.compareTo(p.f) != 0) {
                return f.compareTo(p.f);
            }
            return s.compareTo(p.s);
        }

        @Override
        public int hashCode() {
            return f.hashCode() ^ s.hashCode();
        }

        @Override
        public boolean equals(Object o) {
            if (this == o) {
                return true;
            }
            if (o == null || this.f == null || this.s == null) {
                return false;
            }
            if (this.getClass() != o.getClass()) {
                return false;
            }
            Pair p = (Pair) o;
            return this.f.equals(p.f) && this.s.equals(p.s);
        }

        @Override
        public String toString() {
            return "{" + f.toString() + ", " + s.toString() + "}";
        }
    }


    public static int nCk(int n, int k, int M) {
        long ret = 1;
        int min = Math.min(k, n - k);
        for (int i = 1; i <= min; i++) {
            ret = (ret * pow(i, M - 2, M)) % M;
        }
        for (int i = n - min + 1; i <= n; i++) {
            ret = (ret * i) % M;
        }
        return (int) ret;
    }

    //return a^b mod M O(logB)
    public static long pow(long a, long b, int M) {
        long ret = 1;
        long tmp = a;
        while (b > 0) {
            if ((b & 1) == 1) ret = (ret * tmp) % M;
            tmp = (tmp * tmp) % M;
            b = b >> 1;
        }
        return ret;
    }


    public static int find(int[] tree, int idx) {
        if (tree[idx] == idx) return idx;
        else return tree[idx] = find(tree, tree[idx]);
    }

    public static void union(int[] tree, int idx1, int idx2) {
        int root1 = find(tree, idx1);
        int root2 = find(tree, idx2);
        tree[root2] = root1;
    }

    public static int lowerBound(int[] list, int target) {
        int ok = list.length;
        int ng = -1;
        while (Math.abs(ok - ng) > 1) {
            int mid = (ok + ng) / 2;
            if (list[mid] >= target) {
                ok = mid;
            } else {
                ng = mid;
            }
        }
        return ok;
    }

    public static int lowerBound(long[] list, long target) {
        int ok = list.length;
        int ng = -1;
        while (Math.abs(ok - ng) > 1) {
            int mid = (ok + ng) / 2;
            if (list[mid] >= target) {
                ok = mid;
            } else {
                ng = mid;
            }
        }
        return ok;
    }

    public static <T extends Comparable<? super T>> int lowerBound(List<T> list, T target) {
        return ~Collections.binarySearch(list, target, (x, y) -> (x.compareTo(y) >= 0) ? 1 : -1);
    }

    public static <T extends Comparable<? super T>> int upperBound(List<T> list, T target) {
        return ~Collections.binarySearch(list, target, (x, y) -> (x.compareTo(y) > 0) ? 1 : -1);
    }
}


class FastScanner {
    private final InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

    private boolean hasNextByte() {
        if (ptr < buflen) {
            return true;
        } else {
            ptr = 0;
            try {
                buflen = in.read(buffer);
            } catch (IOException e) {
                e.printStackTrace();
            }
            if (buflen <= 0) {
                return false;
            }
        }
        return true;
    }

    private int readByte() {
        if (hasNextByte()) return buffer[ptr++];
        else return -1;
    }

    private boolean isPrintableChar(int c) {
        return 33 <= c && c <= 126;
    }

    public boolean hasNext() {
        while (hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;
        return hasNextByte();
    }

    public String next() {
        if (!hasNext()) throw new NoSuchElementException();
        StringBuilder sb = new StringBuilder();
        int b = readByte();
        while (isPrintableChar(b)) {
            sb.appendCodePoint(b);
            b = readByte();
        }
        return sb.toString();
    }

    public long nextLong() {
        if (!hasNext()) throw new NoSuchElementException();
        long n = 0;
        boolean minus = false;
        int b = readByte();
        if (b == '-') {
            minus = true;
            b = readByte();
        }
        if (b < '0' || '9' < b) {
            throw new NumberFormatException();
        }
        while (true) {
            if ('0' <= b && b <= '9') {
                n *= 10;
                n += b - '0';
            } else if (b == -1 || !isPrintableChar(b)) {
                return minus ? -n : n;
            } else {
                throw new NumberFormatException();
            }
            b = readByte();
        }
    }

    public int nextInt() {
        long nl = nextLong();
        if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) throw new NumberFormatException();
        return (int) nl;
    }

    public double nextDouble() {
        return Double.parseDouble(next());
    }
}

class BIT<T> {
    int n;
    ArrayList<T> bit;
    BiFunction<T, T, T> bif;

    /**
     * 1-indexed なBinary Indexed Treeを構築する
     *
     * @param n   容量
     * @param bif 適用させる関数
     * @param sup 初期値
     */
    BIT(int n, BiFunction<T, T, T> bif, Supplier<T> sup) {
        this.n = n;
        bit = new ArrayList<>(n + 1);
        for (int i = 0; i < n + 1; ++i) {
            bit.add(sup.get());
        }
        this.bif = bif;
    }

    /**
     * iの位置の値をvで更新する
     *
     * @param i index
     * @param v 新しい値
     */
    void set(int i, T v) {
        for (int x = i; x <= n; x += x & -x) {
            bit.set(x, bif.apply(bit.get(x), v));
        }
    }

    /**
     * クエリー
     *
     * @param defaultValue 初期値
     * @param i            index
     * @return [1, i]までfを適用した結果
     */
    T reduce(T defaultValue, int i) {
        T ret = defaultValue;
        for (int x = i; x > 0; x -= x & -x) {
            ret = bif.apply(ret, bit.get(x));
        }
        return ret;
    }
}

class SegmentTree<T> {
    int n;
    ArrayList<T> dat;
    BiFunction<T, T, T> bif;
    Supplier<T> sup;

    /**
     * 0-indexed なSegment Treeを構築する
     *
     * @param n_  要求容量
     * @param bif 適用させる関数
     * @param sup 初期値
     */
    SegmentTree(int n_, BiFunction<T, T, T> bif, Supplier<T> sup) {
        n = 1;
        while (n < n_) n *= 2;

        dat = new ArrayList<>(2 * n - 1);
        for (int i = 0; i < 2 * n - 1; ++i) {
            dat.add(sup.get());
        }
        this.bif = bif;
        this.sup = sup;
    }

    /**
     * kの位置の値をvで更新する
     *
     * @param k index
     * @param v 新しい値
     */
    void set(int k, T v) {
        k += n - 1;
        dat.set(k, v);
        while (k > 0) {
            k = (k - 1) / 2;
            dat.set(k, bif.apply(dat.get(k * 2 + 1), dat.get(k * 2 + 2)));
        }
    }

    /**
     * クエリー
     *
     * @param l はじめ
     * @param r おわり
     * @return [l, r)での演算bifを適用した結果を返す
     */
    T reduce(int l, int r) {
        return _reduce(l, r, 0, 0, n);
    }

    T _reduce(int a, int b, int k, int l, int r) {
        if (r <= a || b <= l) return sup.get();
        if (a <= l && r <= b) return dat.get(k);
        T vl = _reduce(a, b, k * 2 + 1, l, (l + r) / 2);
        T vr = _reduce(a, b, k * 2 + 2, (l + r) / 2, r);
        return bif.apply(vl, vr);
    }
}

class UnionFind {
    int[] par;

    UnionFind(int n) {
        par = new int[n];
        for (int i = 0; i < n; ++i) {
            par[i] = i;
        }
    }

    int find(int x) {
        if (par[x] == x) {
            return x;
        }
        return par[x] = find(par[x]);
    }

    boolean same(int x, int y) {
        return find(x) == find(y);
    }

    void union(int x, int y) {
        x = find(x);
        y = find(y);
        if (x == y) {
            return;
        }
        par[x] = y;
    }
}