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 C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Diagnostics.CodeAnalysis;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Runtime.Serialization.Formatters.Binary;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading;
using System.Threading.Tasks;

static class MainClass
{
    private static Scanner scanner;
    public static void Main()
    {
        scanner = new Scanner();
        Solve();
        // for scanner bug
        scanner.Next();
    }

    private static void Solve()
    {
        //1344
        var N = scanner.NextInt();
        var S = scanner.Next();
        var startCount = S.Count(x => x == '(');
        var endCount = S.Count(x => x == ')');
        var beforeStartParen = 0;
        var removed = 0;
        for (int i = 0; i < N; i++)
        {
            if (S[i] == '(')
            {
                beforeStartParen++;
            }
            else
            {
                if (beforeStartParen > 0)
                {
                    removed++;
                    beforeStartParen--;
                }
            }
        }

        startCount -= removed;
        endCount -= removed;

        var sb = new StringBuilder();
        sb.Append(S);
        sb.Insert(0, Enumerable.Repeat('(', endCount).IEnumerableListToString());
        sb.Append(new string(Enumerable.Repeat(')', startCount).ToArray()));

        Console.WriteLine(sb.ToString());
    }


    private static int UpperBound<T>(this IEnumerable<T> list, T item) where T : IComparable<T> => UpperBound(list.ToList(), item);
    private static int LowerBound<T>(this IEnumerable<T> list, T item) where T : IComparable<T> => LowerBound(list.ToList(), item);

    private static int UpperBound<T>(this List<T> list, T item) where T : IComparable<T>
    {
        return list.BinarySearch(item, -1, list.Count, (first, second) => first.CompareTo(second) > 0);
    }

    private static int LowerBound<T>(this List<T> list, T item) where T : IComparable<T>
    {
        return list.BinarySearch(item, -1, list.Count, (first, second) => first.CompareTo(second) >= 0);
    }

    private static int BinarySearch<T>(this List<T> list, T item, int left, int right, Func<T, T, bool> isOkPredicate)
    {
        while (Math.Abs(right - left) > 1)
        {
            var mid = left + (right - left) / 2;
            if (isOkPredicate(list[mid], item))
            {
                right = mid;
            }
            else
            {
                left = mid;
            }
        }
        return right;
    }


    #region ライブラリ

    public static long ToInt64(this string str) => long.Parse(str);
    public static bool IsInt32(this string str)
    {
        var o = 0;
        return int.TryParse(str, out o);
    }
    public static int ToInt32(this string str) => str == "" ? 0 : int.Parse(str);
    public static double ToDouble(this string str) => double.Parse(str);
    public static BigInteger ToBigInteger(this string str) => BigInteger.Parse(str);
    public static List<string> SplittedStringToList(this string str) => str.Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries).ToList();
    public static List<int> SplittedStringToInt32List(this string str) => str.Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries).Select(x => int.Parse(x)).ToList();
    public static List<long> SplittedStringToInt64List(this string str) => str.Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries).Select(x => long.Parse(x)).ToList();
    public static List<BigInteger> SplittedStringToBigInteger(this string str) => str.Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries).Select(x => BigInteger.Parse(x)).ToList();
    public const int INF = int.MaxValue / 4;
    public const long LONGINF = long.MaxValue / 2;
    public const int Mod1e9 = 1000000007;
    public static List<int> ToCumulativeSumList(this List<int> list)
    {
        var ls = new List<int>
        {
            list[0]
        };
        var count = list.Count;
        for (int i = 1; i < count; i++)
        {
            ls.Add(list[i] + ls.Last());
        }
        return ls;
    }

    public static void PrintArray(int[,] array)
    {
        for (int i = 0; i < array.GetLength(0); i++)
        {
            var sb = new StringBuilder();
            for (int j = 0; j < array.GetLength(1); j++)
            {
                if (array[i, j] < 0)
                {
                    sb.Append(-1).Append(" ");
                }
                else
                {
                    sb.Append(array[i, j]).Append(" ");
                }
            }
            Console.WriteLine(sb.ToString().Trim());
        }
    }
    public static int Manhattan(int x1, int x2, int y1, int y2)
    {
        return Math.Abs(x1 - x2) + Math.Abs(y1 - y2);
    }

    public static int Manhattan(KeyValuePair<int, int> keyValuePair1, KeyValuePair<int, int> keyValuePair2)
    {
        return Manhattan(keyValuePair1.Key, keyValuePair2.Key, keyValuePair1.Value, keyValuePair2.Value);
    }

    public static T DeepClone<T>(this T src)
    {
        var memoryStream = new MemoryStream();
        var binaryFormatter = new BinaryFormatter();
        binaryFormatter.Serialize(memoryStream, src);
        memoryStream.Seek(0, SeekOrigin.Begin);
        memoryStream.Dispose();

        return (T)binaryFormatter.Deserialize(memoryStream);
    }

    public static Dictionary<T, int> ToCountedDictionary<T>(this IEnumerable<T> items)
    {
        var dic = new Dictionary<T, int>();
        foreach (var item in items)
        {
            if (dic.ContainsKey(item))
                dic[item]++;
            else
                dic.Add(item, 1);
        }
        return dic;
    }
    public static Dictionary<TKey, TValue> KeyValuePairListToDictionary<TKey, TValue>(this IEnumerable<KeyValuePair<TKey, TValue>> pairs)
    {
        var dic = new Dictionary<TKey, TValue>();
        foreach (var item in pairs)
        {
            dic.Add(item.Key, item.Value);
        }
        return dic;
    }
    public class Scanner
    {
        private readonly TextReader reader;
        public Scanner()
        {
            reader = Console.In;
        }
        public Scanner(TextReader reader)
        {
            this.reader = reader;
        }
        private string NextWord()
        {
            var sb = new StringBuilder();
            while (true)
            {
                var c = reader.Read();
                if ((c == 13 || c == 10 || c == 32) && sb.Length > 0 || c == -1)
                {
                    break;
                }
                sb.Append((char)c);
            }
            return sb.ToString().Trim();
        }
        public int NextInt() => NextWord().ToInt32();
        public string Next() => NextWord();
        public double NextDouble() => NextWord().ToDouble();
        public long NextLong() => NextWord().ToInt64();
        public BigInteger NextBigInteger() => NextWord().ToBigInteger();
        public string NextLine()
        {
            var str = reader.ReadLine();
            if (string.IsNullOrWhiteSpace(str))
            {
                return reader.ReadLine();
            }
            else
            {
                return str;
            }
        }
        private List<string> NextLineToList() => NextLine().Split(new char[] { ' ' },
                                                                  StringSplitOptions.RemoveEmptyEntries)
                                                           .ToList();
        public List<int> NextLineToIntList() => NextLineToList().Select(x => int.Parse(x))
                                                                .ToList();
    }
    public static long Sums(IEnumerable<int> list)
    {
        var sum = 0L;
        foreach (var item in list)
        {
            sum += Math.Abs(item);
        }
        return sum;
    }
    public static string IEnumerableListToString<T>(this IEnumerable<T> ls, bool withSpace = false)
    {
        var sb = new StringBuilder();
        foreach (var item in ls)
        {
            sb.Append(item);
            if (withSpace)
            {
                sb.Append(" ");
            }
        }
        return sb.ToString();
    }
    #endregion

    public static int[] dx = { -1, 1, 0, 0 };
    public static int[] dy = { 0, 0, 1, -1 };

    public static int[] dxx = { 0, 1, 1, 1, 0, -1, -1, -1 };
    public static int[] dyy = { 1, 1, 0, -1, -1, -1, 0, 1 };

}