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.IOException;
import java.io.InputStream;
import java.io.PrintStream;
import java.util.NoSuchElementException;
public class Main {
	private static final PrintStream ps = System.out;
	private static final InputStream IS = System.in;
	private static final byte[] BUFFER = new byte[1024];
	private static int ptr = 0;
	private static int buflen = 0;

	public static void main(String[] args) {
		int n = ni();
		char[] ch = n().toCharArray();
		
		int lc = 0;
		int rc = 0;
		for (int i = 0; i < n; i++) {
			if (ch[i] == ')') {
				if (rc == 0)
					lc++;
				else
					rc--;
			} else {
				rc++;
			}
		}
		
		for (int i = 0; i < lc; i++) {
			ps.print('(');
		}
		for (int i = 0; i < n; i++) {
			ps.print(ch[i]);
		}
		for (int i = 0; i < rc; i++) {
			ps.print(')');
		}
	}


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

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

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

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

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

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

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

	private static int[] nia(int n) {
		int[] a = new int[n];
		for (int i = 0; i < n; i++)
			a[i] = ni();
		return a;
	}
}