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)

mod reader {
    use std::fmt::Debug;
    use std::str::FromStr;
    use std::collections::VecDeque;
    use std::io::{self, Read};

    fn _read_line() -> String {
        let mut buf = String::new();
        io::stdin().read_to_string(&mut buf);
        buf
    }

    fn _read_vec_deque_string() -> VecDeque<String> {
        _read_line().split_whitespace().map(|s| s.to_string()).collect()
    }

    pub struct Reader {
        buf: VecDeque<String>
    }

    impl Reader {
        pub fn new() -> Reader {
            Reader {buf: VecDeque::new()}
        }

        fn update_buf(&mut self) {
            if self.buf.is_empty() {
                self.buf = _read_vec_deque_string();
            }
        }

        pub fn next<T>(&mut self) -> T where T: FromStr, T::Err: Debug {
            self.update_buf();
            self.buf.pop_front().unwrap().parse::<T>().unwrap()
        }

        pub fn nextN<T>(&mut self, n: usize) -> Vec<T> where T: FromStr, T::Err: Debug {
            let mut vec: Vec<T> = Vec::with_capacity(n);
            for i in 0..n {
                vec.push(self.next());
            }
            vec
        }
    }
}

mod data {
    pub struct UnionFind {
        root : Vec<usize>
    }

    impl UnionFind {
        pub fn new(size : usize) -> UnionFind {
            UnionFind {root : (0..size).collect()}
        }

        pub fn find(&mut self, x : usize) -> usize {
            let xroot = self.root[x];
            if x != xroot {
                self.root[x] = self.find(xroot);
            }
            self.root[x]
        }

        pub fn unite(&mut self, x : usize, y : usize) {
            let xroot = self.find(x);
            self.root[xroot] = self.find(y);
        }

        pub fn same(&mut self, x : usize, y : usize) -> bool {
            self.find(x) == self.find(y)
        }
    }
}

fn with_large_stack(func : fn() -> ()) {
    use std::thread::Builder;
    let builder = Builder::new();
    let th = builder.stack_size(10000000);

    th.spawn(move || {
        func();
    }).unwrap().join();
}

fn main() {
    with_large_stack(_main);
}

fn f(s: String, open: char) -> usize {
    let mut nest = 0;
    let mut cnt = 0;
    for c in s.chars() {
        if c == open {
            nest += 1;
        } else if nest > 0{
            nest -= 1;
        } else {
            cnt += 1;
        }
    }    
    cnt
}

fn _main() {
    let mut reader = reader::Reader::new();

    let _: i32 = reader.next();
    let s: String = reader.next();
    let pre = f(s.clone(), '(');
    let suf = f(s.clone().chars().rev().collect::<String>(), ')');
    println!("{}{}{}", std::iter::repeat("(").take(pre).collect::<String>(), s, std::iter::repeat(")").take(suf).collect::<String>());
}