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 Go (1.6) (judge verdict: Wrong Answer)

package main

import (
	"bufio"
	"fmt"
	"os"
	"strings"
)

func main() {
	r := bufio.NewReader(os.Stdin)
	var n int
	var s string
	fmt.Fscan(r, &n)
	fmt.Fscan(r, &s)

	var ans []string
	opCount := 0
	for i := 0; i < n; i++ {
		fmt.Println(opCount)
		if s[i] == '(' {
			ans = append(ans, "(")
			opCount++
		} else {
			if opCount > 0 {
				// 直前の ( に対応する場合
				opCount--
			} else {
				// 直前に対応する ( が存在しない場合、先頭に ( を挿入
				ans = append([]string{"("}, ans...)
			}
			ans = append(ans, ")")
		}
	}

	for opCount > 0 {
		ans = append(ans, ")")
		opCount--
	}

	fmt.Println(strings.Join(ans, ""))
}

// Utility
func min(x, y int) int {
	if x < y {
		return x
	}
	return y
}

func max(x, y int) int {
	if x > y {
		return x
	}
	return y
}