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 Rust (1.15.1) (judge verdict: Accepted)

#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}

pub trait BinarySearch<T> {
    fn lower_bound(&self, &T) -> usize;
}

impl<T: Ord> BinarySearch<T> for [T] {
    fn lower_bound(&self, x: &T) -> usize {
        let mut low = 0;
        let mut high = self.len();

        while low != high {
            let mid = (low + high) / 2;
            match self[mid].cmp(x) {
                Ordering::Less => {
                    low = mid + 1;
                }
                Ordering::Equal | Ordering::Greater => {
                    high = mid;
                }
            }
        }
        low
    }
}

fn main() {
    let n = read::<usize>();
    let s: String = read();
    let s = s.chars().collect::<Vec<_>>();

    let mut right_bracket_poses = Vec::new();
    let mut left_bracket_poses = Vec::new();

    for (i, &ch) in s.iter().enumerate() {
        // println!("{}", ch);
        if ch == '(' {
            left_bracket_poses.push(i);
        } else {
            right_bracket_poses.push(i);
        }
    }

    let mut not_corresspond_left = 0;
    let mut right_for_search = right_bracket_poses.clone();
    for &pos in left_bracket_poses.iter().rev() {
        let cor_pos = right_for_search.lower_bound(&pos);
        if cor_pos == right_for_search.len() {
            not_corresspond_left += 1;
        } else {
            right_for_search.remove(cor_pos);
        }
    }
    // println!("{}", not_corresspond_left);

    let mut not_corresspond_right = 0;
    let mut left_for_search = left_bracket_poses
        .iter()
        .rev()
        .map(|&x| n - 1 - x)
        .collect::<Vec<_>>();
    for &pos in right_bracket_poses.iter() {
        let cor_pos = left_for_search.lower_bound(&(n - 1 - pos));
        if cor_pos == left_for_search.len() {
            not_corresspond_right += 1;
        } else {
            left_for_search.remove(cor_pos);
        }
    }
    // println!("{}", not_corresspond_right);

    let mut s = s.into_iter().collect::<VecDeque<_>>();
    for i in 0..not_corresspond_right {
        s.push_front('(');
    }
    for i in 0..not_corresspond_left {
        s.push_back(')');
    }
    for ch in s {
        print!("{}", ch);
    }
    println!("");
}