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 (1.42.0) (judge verdict: Wrong Answer)

// // -*- coding:utf-8-unix -*-

// #[macro_use]
// extern crate lazy_static;
// extern crate num_bigint; // 0.2.2
// extern crate num_traits; // 0.2.8
// use num_bigint::BigInt;
// use num_traits::Pow;
// use petgraph::algo::dijkstra;
// use petgraph::Directed;

// // use proconio::derive_readable;
// use proconio::fastout;
// use proconio::input;
// // use std::convert::TryInto;
// use ascii::{AsciiChar, AsciiString};
// use itertools::concat;
// use lazy_static::lazy_static;
// use libm::*;
// use petgraph::Graph;
// use std::cmp::*;
// use std::collections::{BinaryHeap, HashMap, HashSet, VecDeque};
// use std::io::*;
// use std::ops::Range;
// use std::str::FromStr;
// use std::sync::Mutex;
// use superslice::*;

// pub fn read<T: FromStr>() -> T {
//     let stdin = stdin();
//     let stdin = stdin.lock();
//     let token: String = stdin
//         .bytes()
//         .map(|c| c.expect("failed to read char") as char)
//         .skip_while(|c| c.is_whitespace())
//         .take_while(|c| !c.is_whitespace())
//         .collect();
//     token.parse().ok().expect("failed to parse token")
// }

// // lazy_static! {
// //     static ref H: Mutex<Vec<i32>> = Mutex::default();
// //     static ref W: Mutex<Vec<i32>> = Mutex::default();
// // }

// // let mut values = VALUES.lock().unwrap();
// // values.extend_from_slice(&[1, 2, 3, 4]);
// // assert_eq!(&*values, &[1, 2, 3, 4]);

use proconio::{input, marker::Chars};
use std::collections::HashMap;

// static ASCII_LOWER: [char; 26] = [
//     'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's',
//     't', 'u', 'v', 'w', 'x', 'y', 'z',
// ];

// abc064-D
fn main() {
    input! {
        n: usize,
        s: Chars
    }

    let mut from_l = vec![-1; n];
    let mut from_r = vec![-1; n];

    from_l[0] = if s[0] == ')' { -1 } else { 1 };
    for (i, s) in s.iter().enumerate() {
        if (i == 0) {
            continue;
        }

        from_l[i] = from_l[i - 1] + if *s == ')' { -1 } else { 1 };
    }

    from_r[0] = if *s.last().unwrap() == '(' { -1 } else { 1 };
    for (i, s) in s.iter().rev().enumerate() {
        if (i == 0) {
            continue;
        }

        from_r[i] = from_r[i - 1] + if *s == '(' { -1 } else { 1 };
    }

    let to_fill_l = from_l.iter().min().unwrap() * -1;
    let to_fill_r = from_r.last().unwrap() * -1 + to_fill_l;
    eprintln!("{},{}", to_fill_l, to_fill_r);

    for _ in 0..to_fill_l {
        print!("(");
    }
    for s in s.iter() {
        print!("{}", s);
    }
    for _ in 0..to_fill_r {
        print!(")");
    }
    println!("");

    return;
}