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 Kotlin (1.0.0) (judge verdict: Accepted)

import java.util.*
import kotlin.*

import java.io.InputStream
import java.lang.NumberFormatException
import java.lang.StringBuilder


fun main(args: Array<String>) {
    Program().solve()
}

class Program {
    fun solve() {
        val sc = FastScanner()
        val n = sc.nextInt()
        val s = sc.next()
        // '('があったら+1 ')'があったら-1する
        // 括弧列の性質
        // 全体では0になる
        // 最小値は0

        // マイナスになったら先頭に'('を追加
        // 最後まで見たとき0より大きかったら')'を追加

        var front = 0
        var tmp = 0
        for (i in 0 until n) {
            when (s[i]) {
                '(' -> tmp++
                ')' -> tmp--
            }
            if (tmp < 0) {
                front++
                tmp++
            }
        }

        println("${String(CharArray(front) { '(' })}$s${String(CharArray(tmp) { ')' })}")
    }


    class FastScanner {
        companion object {
            val input: InputStream = System.`in`
            val buffer = ByteArray(1024) { 0 }
            fun isPrintableChar(c: Int): Boolean = c in 33..126
        }

        var ptr = 0
        var buflen = 0
        private fun hasNextByte(): Boolean {
            if (ptr < buflen) {
                return true
            } else {
                ptr = 0
                buflen = input.read(buffer)
                if (buflen <= 0) {
                    return false
                }
            }
            return true
        }

        private fun readByte(): Int = if (hasNextByte()) buffer[ptr++].toInt() else -1

        private fun skipUnprintable() {
            while (hasNextByte() && !isPrintableChar(buffer[ptr].toInt())) ptr++
        }

        fun hasNext(): Boolean {
            skipUnprintable()
            return hasNextByte()
        }

        fun next(): String {
            if (!hasNext()) throw NoSuchElementException()
            val sb = StringBuilder()
            var b = readByte()
            while (isPrintableChar(b)) {
                sb.appendCodePoint(b)
                b = readByte()
            }
            return sb.toString()
        }

        fun nextInt(): Int {
            if (!hasNext()) throw NoSuchElementException()
            var n = 0
            var b = readByte()
            // '-' = 45
            val minus = b == 45
            if (minus) {
                b = readByte()
            }

            // '0' = 48 '9' = 57
            if (b !in 48..57) {
                throw NumberFormatException()
            }

            while (true) {
                if (b in 48..57) {
                    n *= 10
                    n += b - 48
                } else if (b == -1 || !isPrintableChar(b)) {
                    return if (minus) -n else n
                } else {
                    throw NumberFormatException()
                }
                b = readByte()
            }
        }

        fun nextLong(): Long {
            if (!hasNext()) throw NoSuchElementException()
            var n = 0L
            var b = readByte()
            // '-' = 45
            val minus = b == 45
            if (minus) {
                b = readByte()
            }

            // '0' = 48 '9' = 57
            if (b !in 48..57) {
                throw NumberFormatException()
            }

            while (true) {
                if (b in 48..57) {
                    n *= 10
                    n += b - 48
                } else if (b == -1 || !isPrintableChar(b)) {
                    return if (minus) -n else n
                } else {
                    throw NumberFormatException()
                }
                b = readByte()
            }
        }

        fun nextDouble(): Double = next().toDouble()
    }
}