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: Wrong Answer)

import java.io.*;
import java.util.*;



class Main{

    void solve(){
        int N = ni();
        String s = ns();
        int[] d = new int[N];
        int left = 0;
        int right = 0;
        for(int i = 0; i < N; i++){
            if(s.charAt(i) == '('){
                left++;
            } else {
                right++;
            }
            d[i] = left - right;
        }
        int x = min(d);

        for(int i = 0; i < x * -1; i++){
            s = "(" + s;
        }
        for(int i = 0; i < d[N - 1] - x; i++){
            s = s + ")";
        }
        
        out.println(s);
        

        out.flush();
    } 


    public static void main(String[] args){
        Main m = new Main();
        m.solve();
    }

    Main(){
        this.scan = new FastScanner();
        this.out = new PrintWriter(System.out);
    }

    private FastScanner scan;
    private PrintWriter out;
    private final int MOD = 1_000_000_007;
    private final int INF = 2_147_483_647;
    private final long LINF = 9223372036854775807L;
    private long[] fac;
    private long[] finv;
    private long[] inv;

    // Scanner
    int ni(){ return scan.nextInt();}
    int[] ni(int n){int[] a = new int[n]; for(int i = 0; i < n; i++){a[i] = ni();} return a;}
    int[][] ni(int y, int x){int[][] a = new int[y][x];
        for(int i = 0; i < y; i++){for(int j = 0; j < x; j++){a[i][j] = ni();}} return a;}
    long nl(){return scan.nextLong();}
    long[] nl(int n){long[] a = new long[n]; for(int i = 0; i < n; i++){a[i] = nl();} return a;}
    long[][] nl(int y, int x){long[][] a = new long[y][x];
        for(int i = 0; i < y; i++){for(int j = 0; j < x; j++){a[i][j] = nl();}} return a;}
    String ns(){return scan.next();}
    String[] ns(int n){String[] a = new String[n]; for(int i = 0; i < n; i++){a[i] = ns();} return a;}
    String[][] ns(int y, int x){String[][] a = new String[y][x];
        for(int i = 0; i < y; i++){for(int j = 0; j < x; j++){a[i][j] = ns();}} return a;}

    // Mathematics
    int max(int a, int b){return Math.max(a, b);}
    long max(long a, long b){return Math.max(a, b);}
    double max(double a, double b){return Math.max(a, b);}
    int max(int[] a){int max = a[0]; for(int value:a){max = max(max,value);} return max;}
    long max(long[] a){long max = a[0]; for(long value:a){max = max(max,value);} return max;}
    double max(double[] a){double max = a[0]; for(double value:a){max = max(max,value);} return max;}
    int min(int a, int b){return Math.min(a, b);}
    long min(long a, long b){return Math.min(a, b);}
    double min(double a, double b){return Math.min(a, b);}
    int min(int[] a){int min = a[0]; for(int value:a){min = min(min,value);} return min;}
    long min(long[] a){long min = a[0]; for(long value:a){min = min(min,value);} return min;}
    double min(double[] a){double min = a[0]; for(double value:a){min = min(min,value);} return min;}
    long sum(int[] a){long sum = 0; for(int value:a){sum += value;} return sum;}
    long sum(long[] a){long sum = 0; for(long value:a){sum += value;} return sum;}
    double sum(double[] a){double sum = 0; for(double value:a){sum += value;} return sum;}
    long mod(long n) { n %= MOD; return n + (n < 0 ? MOD : 0); }
    int gcd(int a, int b){return b == 0 ? a : gcd(b, a % b);}
    long gcd(long a, long b){return b == 0 ? a : gcd(b, a % b);}
    int lcm(int a, int b){return a / gcd(a, b) * b;}
    long lcm(long a, long b){return a / gcd(a, b) * b;}
    long fact(int n){ if(n == 0){ return 1; } long a = n; for(long i = n - 1; i >= 2; i--){ a = a % MOD * i; } return a; }
    long fact(long n){ if(n == 0){ return 1; } long a = n; for(long i = n - 1; i >= 2; i--){ a = a % MOD * i; } return a; }
    // nPr(int)
    long npr(int n, int r){
        long a = 1;
        for(int i = n; i > n - r; i--){
            a *= i;
        }
        return a;
    }
    // nPr(long)
    long npr(long n, long r){
        long a = 1;
        for(long i = n; i > n - r; i--){
            a *= i;
        }
        return a;
    }
    // 素数判定(int)
    boolean checkPrime(int n){
        for(int i = 2; i * i <= n; i++){
            if(n % i == 0){
                return false;
            }
        }
        return true;
    }
    // 素数判定(long)
    boolean checkPrime(long n){
        for(long i = 2; i * i <= n; i++){
            if(n % i == 0){
                return false;
            }
        }
        return true;
    }
    // エラトステネスの篩
    long[] erathos(int n){
        boolean[] flag = new boolean[n + 1];
        TreeSet<Integer> nums = new TreeSet<>();
        for(int i = 2; i <= n; i++){
            if(flag[i]) continue;
            nums.add(i);
            for(int j = i * 2; j <= n; j += i){
                flag[j] = true;
            }
        }
        long[] primes = new long[nums.size()];
        int index = 0;
        for(int num : nums){
            primes[index] = num;
            index++;
        }

        return primes;
    }

    // mod pにおける累乗 a^n
    long modpow(long a, long n, long p){
        long res = 1;
        while(n > 0){
            if((n & 1) == 1){
                res = res * a % p;
            }
            a = a * a % p;
            n = n >> 1;
        }
        return res;
    }
    // mod pにおけるaの逆元a^-1
    long modinv(long a, long p){
        return modpow(a, p - 2, p);
    }
    // fac,finv,invの初期化
    void comInit(int max){
        fac = new long[max];
        finv = new long[max];
        inv = new long[max];
        fac[0] = fac[1] = 1;
        finv[0] = finv[1] = 1;
        inv[1] = 1;
        for(int i = 2; i < max; i++){
            fac[i] = fac[i - 1] * i % MOD;
            inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;
            finv[i] = finv[i - 1] * inv[i] % MOD;
        }
    }
    // 二項係数nCr
    long com(int n, int r){
        if(n < r || (n < 0 || r < 0)){
            return 0;
        }
        return fac[n] * (finv[r] * finv[n - r] % MOD) % MOD;
    }
    // 二項係数nCr（nが10^9など巨大なとき用）
    long ncr(long n, long k){
        long a = 1;
        long b = 1;
        for(int i = 1; i <= k; i++){
            a = a * (n + 1 - i) % MOD;
            b = b * i % MOD;
        }
        
        return modinv(b, MOD) * a % MOD;
    }
    // 二次元上の二点間の距離
    double distance(double x1, double y1, double x2, double y2){
        double dist = Math.sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2));
        return dist;
    }
    // 三次元上の二点間の距離
    double distance(double x1, double y1, double z1, double x2, double y2, double z2){
        double dist = Math.sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2) + (z1 - z2) * (z1 - z2));
        return dist;
    }


    // Array
    void sort(int[] a){ Arrays.sort(a);}
    void sort(long[] a){ Arrays.sort(a);}
    void sort(double[] a){ Arrays.sort(a);}
    void sort(String[] a){ Arrays.sort(a);}
    int[] reverse(int[] a){
        int[] reversed = new int[a.length];
        for(int i = 0; i < a.length; i++){
            reversed[a.length - i - 1] = a[i];
        }
        return reversed;
    }
    long[] reverse(long[] a){
        long[] reversed = new long[a.length];
        for(int i = 0; i < a.length; i++){
            reversed[a.length - i - 1] = a[i];
        }
        return reversed;
    }
    double[] reverse(double[] a){
        double[] reversed = new double[a.length];
        for(int i = 0; i < a.length; i++){
            reversed[a.length - i - 1] = a[i];
        }
        return reversed;
    }
    char[] reverse(char[] a){
        char[] reversed = new char[a.length];
        for(int i = 0; i < a.length; i++){
            reversed[a.length - i - 1] = a[i];
        }
        return reversed;
    }
    String[] reverse(String[] a){
        String[] reversed = new String[a.length];
        for(int i = 0; i < a.length; i++){
            reversed[a.length - i - 1] = a[i];
        }
        return reversed;
    }
    boolean[] reverse(boolean[] a){
        boolean[] reversed = new boolean[a.length];
        for(int i = 0; i < a.length; i++){
            reversed[a.length - i - 1] = a[i];
        }
        return reversed;
    }
    void fill(int[] array, int x) { Arrays.fill(array, x); }
    void fill(long[] array, long x) { Arrays.fill(array, x); }
    void fill(double[] array, double x) { Arrays.fill(array, x); }
    void fill(boolean[] array, boolean x) { Arrays.fill(array, x); }
    void fill(int[][] array, int x) { for(int a[] : array) { fill(a, x); } }
    void fill(long[][] array, long x) { for(long a[] : array) { fill(a, x); } }
    void fill(double[][] array, double x) { for(double a[] : array) { fill(a, x); } }
    void fill(boolean[][] array, boolean x) { for(boolean a[] : array) { fill(a, x); } }
    void fill(int[][][] array, int x) { for(int[][] ary : array) { for(int[] a : ary){ fill(a, x); } } }
    void fill(long[][][] array, long x) { for(long[][] ary : array) { for(long[] a : ary){ fill(a, x); } } }

    // Algorithm
    // 深さ優先探索
    void dfs(Node[] nodes, int v, boolean[] seen){
        seen[v] = true;
        
        for(Edge edge : nodes[v].edges){
            if(seen[edge.to]) continue;
            dfs(nodes, edge.to, seen);
        }

    }
    // 幅優先探索
    /* long[] bfs(Node[] nodes, int start){
        Queue<Integer> queue = new ArrayDeque<>();
        queue.add(start);
        long[] dist = new long[nodes.length];
        fill(dist, -1);
        dist[start] = 0;

        while(!queue.isEmpty()){
            int now = queue.poll();
            for(Edge edge : nodes[now].edges){
                if(dist[edge.to] != -1) continue;
                dist[edge.to] = dist[now] + 1;
                queue.add(edge.to);
            }
        }

        return dist;
    } */
    
    // ダイクストラ法
    long[] dijkstra(Node[] nodes, int start){
        Queue<Edge> queue = new PriorityQueue<>();
        long[] dist = new long[nodes.length];
        fill(dist, LINF / 2);
        dist[start] = 0;
        queue.add(new Edge(start, start, dist[start]));

        while(!queue.isEmpty()){
            Edge now = queue.poll();        
            if(dist[now.to] < now.cost) continue;
            for(Edge edge : nodes[now.to].edges){
                if(dist[edge.to] > dist[edge.from] + edge.cost){
                    dist[edge.to] = dist[edge.from] + edge.cost;
                    queue.add(new Edge(edge.from, edge.to, dist[edge.to]));
                    nodes[edge.to].past = edge.from;
                }
            }
        }

        return dist;
    }
    
    // ソート済みint型配列でkey以上の値の最小indexを返す
    int lowerBound(int[] a, int key){
        int ng = -1;
        int ok = a.length;
        while(Math.abs(ok - ng) > 1){
            int mid = (ok + ng) / 2; 
            if(a[mid] >= key){
                ok = mid;
            } else {
                ng = mid;
            }
        }
        return ok;
    }
    // ソート済みlong型配列でkey以上の値の最小indexを返す
    int lowerBound(long[] a, long key){
        int ng = -1;
        int ok = a.length;
        while(Math.abs(ok - ng) > 1){
            int mid = (ok + ng) / 2; 
            if(a[mid] >= key){
                ok = mid;
            } else {
                ng = mid;
            }
        }
        return ok;
    }
    // 文字列sとtの最長共通部分列の長さを返す
    int lcs(String s , String t){
        int[][] dp = new int[s.length() + 1][t.length() + 1];

        for(int i = 0; i < s.length(); i++){
            for(int j = 0; j < t.length(); j++){
                if(s.charAt(i) == t.charAt(j)){
                    dp[i + 1][j + 1] = max(dp[i][j] + 1, dp[i + 1][j + 1]);
                }
                dp[i + 1][j + 1] = max(dp[i + 1][j + 1], dp[i + 1][j]);
                dp[i + 1][j + 1] = max(dp[i + 1][j + 1], dp[i][j + 1]);
            }
        }

        return dp[s.length()][t.length()];
    }

}

// グリッド型のグラフ
class Grid{
    Node[] nodes;             
    List<Node> starts;  // 特定のノードのみ始点にしたい場合はこのリストに入れる
    String[] field;
    int H;
    int W;
    char symbol1;
    char symbol2;

    Grid(String[] field, int H, int W, char symbol1, char symbol2){
        this.nodes = new Node[H * W + 1];
        this.field = field;
        this.starts = new ArrayList<>();
        this.H = H;
        this.W = W;
        this.symbol1 = symbol1;
        this.symbol2 = symbol2;
    }

    void makeGraph(){
        for(int i = 0; i <= H * W; i++){    // 0はダミーなので注意
            nodes[i] = new Node(i, -1);
        }
        
        int[][] ch = new int[H + 2][W + 2];
        for(int i = 0; i < H; i++){
            for(int j = 0; j < W; j++){
                if(field[i].charAt(j) == symbol1){
                    ch[i + 1][j + 1] = 1;
                } else if(field[i].charAt(j) == symbol2) {
                    ch[i + 1][j + 1] = 2;
                }
            }
        }
        for(int i = 1; i <= H; i++){
            for(int j = 1; j <= W; j++){
                if(ch[i][j] == 1){
                    if(ch[i - 1][j] == 2) nodes[pos(i, j)].addEdge(new Edge(pos(i, j), pos(i - 1, j), 1));
                    if(ch[i + 1][j] == 2) nodes[pos(i, j)].addEdge(new Edge(pos(i, j), pos(i + 1, j), 1));
                    if(ch[i][j - 1] == 2) nodes[pos(i, j)].addEdge(new Edge(pos(i, j), pos(i, j - 1), 1));
                    if(ch[i][j + 1] == 2) nodes[pos(i, j)].addEdge(new Edge(pos(i, j), pos(i, j + 1), 1));
                    starts.add(nodes[pos(i, j)]);    //
                } else if(ch[i][j] == 2){
                    if(ch[i - 1][j] == 2) nodes[pos(i, j)].addEdge(new Edge(pos(i, j), pos(i - 1, j), 1));
                    if(ch[i + 1][j] == 2) nodes[pos(i, j)].addEdge(new Edge(pos(i, j), pos(i + 1, j), 1));
                    if(ch[i][j - 1] == 2) nodes[pos(i, j)].addEdge(new Edge(pos(i, j), pos(i, j - 1), 1));
                    if(ch[i][j + 1] == 2) nodes[pos(i, j)].addEdge(new Edge(pos(i, j), pos(i, j + 1), 1));
                }
            }
        }

    }

    // 座標の言い換えがいちいちめんどくさい！
    int pos(int y, int x){
        return (y - 1) * W + x;
    }

}

// ノード
class Node{
    int id;             // ノード番号
    int past;           // 直前の頂点
    List<Edge> edges;   // 辺のリスト

    Node(int id, int past){
        this.id = id;
        this.past = past;
        this.edges = new ArrayList<>();
    }
    void addEdge(Edge edge){
        edges.add(edge);
    }

    public boolean equals(Object obj){
        if(obj instanceof Node){
            Node node = (Node)obj;
            return this.id == node.id;
        } 
        return false;
    }
    public int hashCode(){
        return id;
    }
    public String toString(){
        return "[id:" + id + " past: " + past + "]";
    }
}

// 辺の情報を持つクラス
class Edge implements Comparable<Edge>{
    int from;       // どの頂点から
    int to;          //  どの頂点へ
    long cost;       // 辺の重み

    Edge(int from, int to, long cost){
        this.from = from;
        this.to = to;
        this.cost = cost;
    }
    public int compareTo(Edge edge){
        return Long.compare(this.cost, edge.cost);
    }

    public String toString(){
        return "[" + from + " to " + to + " cost:" + cost + "]";
    }
}
// 辺の重み比較のComparator
class EdgeComparator implements Comparator<Edge>{
    @Override
    public int compare(Edge e1, Edge e2){
        long cost1 = e1.cost;
        long cost2 = e2.cost;
        if(cost1 < cost2){
            return -1;
        } else if(cost1 > cost2){
            return 1;
        } else {
            return 0;
        }
    }
}
// Union-Find
class UnionFind{
    int[] par;
    int[] size;
    UnionFind(int N){
        par = new int[N];
        size = new int[N];
        for(int i = 0; i < N; i++){
            par[i] = i;
            size[i] = 1;
        }
    }

    void init(int N){
        for(int i = 0; i < N; i++){
            par[i] = i;
            size[i] = 1;
        }
    }

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

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

    void unite(int x, int y){
        x = root(x);
        y = root(y);
        if(x == y) return;
        if(size[x] < size[y]){
            int tmp = x;
            x = y;
            y = tmp;
        }
        size[x] += size[y];
        par[y] = x;
    }

    int size(int x){
        return size[root(x)];
    }

}

// 順列を管理する
class Permutation {
    private int number;
    private int listSize;
    private int searched;
    private int nextIndex;
    private int[][] permList;

    Permutation(int num) {
        this.number = num;
        this.listSize  = this.fact(this.number);
        this.searched   = 0;
        this.nextIndex = 0;
        this.permList  = new int[this.listSize][this.number];

        this.create(0, new int[this.number], new boolean[this.number]);
    }

    int[] nextPerm() {
        return permList[this.nextIndex++];
    }

    boolean isNext() {
        if(this.nextIndex < this.listSize) {
            return true;
        } else {
            this.nextIndex = 0;
            return false;
        }
    }

    int fact(int n){
        return n == 0 ? 1 : n * fact(n-1);
    }

    void create(int num, int[] list, boolean[] flag) {
        if(num == this.number) {
            copyArray(list, permList[this.searched]);
            this.searched++;
        }
        for(int i = 0; i < this.number; i++){
            if(flag[i]) continue;
            list[num] = i;
            flag[i] = true;
            this.create(num+1, list, flag);
            flag[i] = false;

        }
    }

    void copyArray(int[] from, int[] to) {
        for(int i=0; i<from.length; i++) to[i] = from[i];
    }

    void printNum(int[] nums) {
        for(int n : nums) System.out.print(n);
        System.out.println();
    }
}

// 一点更新区間取得Segment Tree
class SegTree{
    int[] dat;
    int size;
    
    SegTree(int N){
        this.dat = new int[N * 4];
        this.size = N;
    }

    // [a, b]の最小値を返す
    // l,rにはノードkに対応する区間を与える
    int query(int a, int b, int k, int l, int r){
        if(r < a || b < l) return Integer.MAX_VALUE;
        if(a <= l && r <= b) return dat[k];
        int vl = query(a, b, k << 1, l, (l + r) / 2);
        int vr = query(a, b, (k << 1) + 1, (l + r) / 2 + 1, r);
        return Math.min(vl, vr);
    }

    // インデックスiの値をxに更新
    void update(int i, int x){
        i += this.size - 1;
        dat[i] = x;
        while(i > 0){
            i = i >> 1;     // 1ビットぶん右シフト
            dat[i] = Math.min(dat[i << 1], dat[(i << 1) + 1]);
        }
    }

    public String toString(){
        StringBuilder sb = new StringBuilder("[");
        for(int i = size; i < 2 * size; i++){
            sb.append(dat[i]).append(",");
        }
        sb.delete(sb.length() - 1, sb.length());
        String output = sb.append("]").toString();
        return output;
    }
}

// Range Sum Queryを実現するBinary Indexed Tree
class BIT{
    int[] tree;

    BIT(int N){
        this.tree = new int[N + 1];
    }

    int sum(int i){
        int sum = 0;
        while(i > 0){
            sum += this.tree[i];
            i -= i & -i;
        }
        return sum;
    }

    void add(int i, int x){
        while(i <= tree.length){
            this.tree[i] += x;
            i += i & -i;
        }
    }

    public String toString(){
        StringBuilder sb = new StringBuilder("[");
        for(int i = 0; i < tree.length; i++){
            sb.append(tree[i] + ",");
        }
        sb.append("]");
        String output = sb.toString();
        return output;
    }
    
}

// 標準のScannerより高速に標準入力する
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 static 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());}
}