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::{max, min, Ordering};
#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::io::{stderr, stdin, stdout, BufWriter, StdoutLock, Write};
#[allow(unused_imports)]
use std::iter::FromIterator;
#[allow(unused_imports)]
use std::{i64, u64, usize};
#[allow(unused_macros)]
macro_rules ! input { ( source = $ s : expr , $ ( $ r : tt ) * ) => { let mut iter = $ s . split_whitespace ( ) ; let mut next = || { iter . next ( ) . unwrap ( ) } ; input_inner ! { next , $ ( $ r ) * } } ; ( $ ( $ r : tt ) * ) => { let stdin = std :: io :: stdin ( ) ; let mut bytes = std :: io :: Read :: bytes ( std :: io :: BufReader :: new ( stdin . lock ( ) ) ) ; let mut next = move || -> String { bytes . by_ref ( ) . map ( | r | r . unwrap ( ) as char ) . skip_while ( | c | c . is_whitespace ( ) ) . take_while ( | c |! c . is_whitespace ( ) ) . collect ( ) } ; input_inner ! { next , $ ( $ r ) * } } ; }
#[allow(unused_macros)]
macro_rules ! input_inner { ( $ next : expr ) => { } ; ( $ next : expr , ) => { } ; ( $ next : expr , $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let $ var = read_value ! ( $ next , $ t ) ; input_inner ! { $ next $ ( $ r ) * } } ; ( $ next : expr , mut $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let mut $ var = read_value ! ( $ next , $ t ) ; input_inner ! { $ next $ ( $ r ) * } } ; }
#[allow(unused_macros)]
macro_rules ! read_value { ( $ next : expr , ( $ ( $ t : tt ) ,* ) ) => { ( $ ( read_value ! ( $ next , $ t ) ) ,* ) } ; ( $ next : expr , [ $ t : tt ; $ len : expr ] ) => { ( 0 ..$ len ) . map ( | _ | read_value ! ( $ next , $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ next : expr , [ $ t : tt ] ) => { { let len = read_value ! ( $ next , usize ) ; ( 0 .. len ) . map ( | _ | read_value ! ( $ next , $ t ) ) . collect ::< Vec < _ >> ( ) } } ; ( $ next : expr , chars ) => { read_value ! ( $ next , String ) . chars ( ) . collect ::< Vec < char >> ( ) } ; ( $ next : expr , bytes ) => { read_value ! ( $ next , String ) . into_bytes ( ) } ; ( $ next : expr , usize1 ) => { read_value ! ( $ next , usize ) - 1 } ; ( $ next : expr , $ t : ty ) => { $ next ( ) . parse ::<$ t > ( ) . expect ( "Parse error" ) } ; }
#[allow(dead_code)]
struct Writer {
    s: String,
}
#[allow(unused_imports)]
use std::fmt::Display;
#[allow(dead_code)]
#[doc = " let mut writer = Writer::new();"]
#[doc = " writer.writeln(hoge);"]
#[doc = " writer.flush()"]
impl Writer {
    #[allow(dead_code)]
    pub fn new() -> Writer {
        Writer { s: String::new() }
    }
    #[allow(dead_code)]
    pub fn flush(&mut self) {
        print!("{}", self.s);
        self.s.clear();
    }
    pub fn write<T: Display>(&mut self, x: T) {
        self.s.push_str(&format!("{}", x));
    }
    pub fn writeln<T: Display>(&mut self, x: T) {
        self.s.push_str(&format!("{}", x));
        self.s.push('\n');
    }
    #[allow(dead_code)]
    pub fn write_vec<T: Display>(&mut self, xs: &Vec<T>) {
        if xs.len() == 0 {
            self.writeln("");
            return;
        }
        self.write(&format!("{}", xs[0]));
        for i in 1..xs.len() {
            self.write(&format!(" {}", xs[i]));
        }
        self.writeln("");
    }
}
#[allow(unused_macros)]
macro_rules ! dbg { ( $ ( $ a : expr ) ,* ) => { writeln ! ( & mut stderr ( ) , concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) . unwrap ( ) ; } }
#[allow(dead_code)]
const INF_U: u64 = 1_000_000_000_000_000;
#[allow(dead_code)]
const INF_I: i64 = 1_000_000_000_000_000;
#[allow(non_snake_case)]
#[allow(dead_code)]
fn main() {
    input! {
        n: usize,
        s: chars
    }
    let mut stack = Vec::new();
    for &ss in &s {
        if !stack.is_empty() {
            if stack.last().unwrap() == &'(' && ss == ')' {
                stack.pop();
                continue;
            }
        }
        stack.push(ss);
    }
    let mut left = 0;
    let mut right = 0;
    for x in stack {
        match x {
            '(' => left += 1,
            ')' => right += 1,
             _  => panic!(),
        }
    }
    for i in 0..right {
        print!("(");
    }
    let a: String = s.into_iter().collect();
    print!("{}", a);
    for i in 0..left {
        print!(")");
    }
    print!("\n");
}