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)

fn main() {
    let mut input_buffer = String::with_capacity(cpio::BUFFER_SIZE);
    let mut output_buffer = Vec::with_capacity(cpio::BUFFER_SIZE);
    unsafe {
        cpio::input::buf = &mut output_buffer;
        cpio::output::buf = &mut input_buffer;
    }
    std::thread::Builder::new()
        .stack_size(104_857_600)
        .spawn(solve)
        .unwrap()
        .join()
        .unwrap();
    cpio::output::flush();
}

#[macro_use]
#[allow(dead_code, non_upper_case_globals, unused_macros)]
mod cpio {
    pub const BUFFER_SIZE: usize = 8192;
    #[macro_use]
    pub mod output {
        pub const ENABLE_DUMP: bool = true;
        pub const PRECISION: usize = 10;
        pub static mut buf: *mut String = 0 as *mut String;

        macro_rules! p {
            ($x: expr) => {{
                $x.output();
                '\n'.output();
                unsafe {
                    if (*cpio::output::buf).len() > cpio::BUFFER_SIZE {
                        flush();
                    }
                }
            }};
            ($x: expr, $($y: expr),+) => {{
                $x.output();
                ' '.output();
                p!($($y),+);
            }};
        }

        macro_rules! pf {
            ($($x: expr),+) => {{
                p!($($x),+);
                flush();
            }};
        }

        macro_rules! d {
            ($($a: expr),+) => {
                if ENABLE_DUMP {
                    use std::io::*;
                    write!(stderr(), "{}:{}\t", file!(), line!()).unwrap();
                    d!(A $($a),+);
                    write!(stderr(), " = ").unwrap();
                    d!(B $($a),+);
                    write!(stderr(), "\n").unwrap();
                    stderr().flush().unwrap();
                }
            };
            (A $x: expr) => {
                write!(stderr(), "{}", stringify!($x)).unwrap();
            };
            (A $x: expr, $($y: expr),+) => {
                write!(stderr(), "{}, ", stringify!($x)).unwrap();
                d!(A $($y),+);
            };
            (B $x: expr) => {
                write!(stderr(), "{:?}", $x).unwrap();
            };
            (B $x: expr, $($y: expr),+) => {
                write!(stderr(), "{:?}, ", $x).unwrap();
                d!(B $($y),+);
            };
        }

        pub trait Output {
            fn output(&self);
        }

        macro_rules! output_normal {
            ($t: ty) => {
                impl Output for $t {
                    fn output(&self) {
                        unsafe {
                            use std::fmt::Write;
                            write!(&mut *buf, "{}", self).unwrap();
                        }
                    }
                }
            };
            ($t: ty, $($u: ty),+) => {
                output_normal!($t);
                output_normal!($($u),+);
            };
        }

        macro_rules! output_float {
            ($t: ty) => {
                impl Output for $t {
                    fn output(&self) {
                        unsafe {
                            use std::fmt::Write;
                            write!(&mut *buf, "{:.*}", PRECISION, self).unwrap();
                        }
                    }
                }
            };
            ($t: ty, $($u: ty),+) => {
                output_float!($t);
                output_float!($($u),+);
            };
        }

        pub fn flush() {
            unsafe {
                print!("{}", *buf);
                use std::io::*;
                stdout().flush().unwrap();
                (*buf).clear();
            }
        }

        output_normal!(u8, u16, u32, u64, usize, char);
        output_normal!(i8, i16, i32, i64, isize);
        output_normal!(bool, &'static str, String);
        output_float!(f32, f64);
    }

    pub mod input {
        pub trait Input<T> {
            fn input() -> T;
        }

        macro_rules! input_primitive {
            ($t: ty) => {
                impl Input<$t> for $t {
                    fn input() -> $t {
                        get_word().expect("EOF?").parse()
                            .unwrap_or_else(|e| panic!("Parse error : {}", e))
                    }
                }
            };
            ($t: ty, $($u: ty),+) => {
                input_primitive!($t);
                input_primitive!($($u),+);
            };
        }

        macro_rules! input_tuple {
            ($($t: ident),*) => {
                impl< $($t: Input<$t>),* > Input< ( $($t),* ) > for ( $($t),* ) {
                    fn input() -> ( $($t),* ) {
                        ( $( $t::input()),* )
                    }
                }
            };
        }

        input_primitive!(u8, u16, u32, u64, usize);
        input_primitive!(i8, i16, i32, i64, isize);
        input_primitive!(bool, String);
        input_tuple!(A, B);
        input_tuple!(A, B, C);
        input_tuple!(A, B, C, D);

        pub fn get<T: Input<T>>() -> T {
            T::input()
        }

        pub fn get_vec<T: Input<T>>(n: usize) -> Vec<T> {
            (0..n).map(|_| get()).collect()
        }

        pub fn get_mat<T: Input<T>>(r: usize, c: usize) -> Vec<Vec<T>> {
            (0..r).map(|_| get_vec(c)).collect()
        }

        pub fn get_vec_char() -> Vec<char> {
            get_word().unwrap().chars().collect()
        }

        pub fn get_mat_char(h: usize) -> Vec<Vec<char>> {
            (0..h).map(|_| get_vec_char()).collect()
        }

        pub fn get_line() -> String {
            get_line_wrapped().unwrap()
        }

        fn get_word() -> Option<String> {
            let mut res = String::with_capacity(18);
            while let Some(c) = get_u8() {
                let d = c as char;
                if !d.is_whitespace() {
                    res.push(d);
                } else if res.len() != 0 {
                    unget_u8(c);
                    break;
                }
            }
            if res.len() == 0 { None } else { Some(res) }
        }

        fn get_line_wrapped() -> Option<String> {
            let c = get_u8();
            if c.is_none() {
                return None;
            }
            let mut line = String::with_capacity(18);
            line.push(c.unwrap() as char);
            loop {
                let c = get_u8();
                if c.is_none() || c.unwrap() == b'\n' {
                    // コメントはC++等での仕様
                    // if c.is_some() {
                    //     self.unget_u8(b'\n');
                    // }
                    return Some(line);
                }
                line.push(c.unwrap() as char);
            }
        }

        pub fn has_next() -> bool {
            loop {
                let c = get_u8();
                if c.is_none() {
                    return false;
                }
                let c = c.unwrap();
                if !(c as char).is_whitespace() {
                    unget_u8(c);
                    return true;
                }
            }
        }

        pub static mut buf: *mut Vec<u8> = 0 as *mut Vec<u8>;

        fn get_u8() -> Option<u8> {
            unsafe {
                let b = &mut (*buf);
                use std::io::{stdin, Read};
                if let Some(c) = b.pop() {
                    Some(c as u8)
                } else {
                    b.resize(super::BUFFER_SIZE, 0);
                    match stdin().read(b) {
                        Ok(l) if l > 0 => {
                            b.truncate(l);
                            b.reverse();
                            b.pop()
                        }
                        _ => {
                            b.clear();
                            return None;
                        }
                    }
                }
            }
        }

        fn unget_u8(c: u8) {
            unsafe { (*buf).push(c) }
        }
    }
}

#[allow(unused_imports)]
use cpio::input::*;
#[allow(unused_imports)]
use cpio::output::*;

fn solve() {

    fn minlast(x: &Vec<char>) -> (i32, i32) {
        let mut d = 0;
        let mut min = 0;
        for &c in x {
            if c == '(' {
                d += 1;
            } else {
                d -= 1;
            }
            if min > d {
                min = d;
            }
        }
        (min, d)
    }

    while has_next() {
        let mut res = String::new();
        let n: usize = get();
        let s = get_vec_char();
        let (mut min, mut last) = minlast(&s);
        while min < 0 {
            res.push('(');
            min += 1;
            last += 1;
        }
        for c in s {
            res.push(c);
        }
        while last > 0 {
            res.push(')');
            last -= 1;
        }
        // use std::iter::*;
        p!(res);
    }
}