Problem:
Score : 600 points

Problem Statement

A bracket sequence is a string that is one of the following:

An empty string;

The concatenation of (, A, and ) in this order, for some bracket sequence A ;

The concatenation of A and B in this order, for some non-empty bracket sequences A and B /

Given are N strings S_i. Can a bracket sequence be formed by concatenating all the N strings in some order?

Constraints

1 \leq N \leq 10^6

The total length of the strings S_i is at most 10^6.

S_i is a non-empty string consisting of ( and ).

Input

Input is given from Standard Input in the following format:

N
S_1
:
S_N

Output

If a bracket sequence can be formed by concatenating all the N strings in some order, print Yes; otherwise, print No.

Sample Input 1

2
)
(()

Sample Output 1

Yes

Concatenating (() and ) in this order forms a bracket sequence.

Sample Input 2

2
)(
()

Sample Output 2

No

Sample Input 3

4
((()))
((((((
))))))
()()()

Sample Output 3

Yes

Sample Input 4

3
(((
)
)

Sample Output 4

No

Solution in C# (.NET Core 3.1.201) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;
using System.Text.RegularExpressions;

namespace AtCoder_Core
{
    class Program
    {
        static long CombinationL(long m, long n)
        {
            if (m == n || n == 0) return 1;

            long ans = 1;
            for (long i = 1; i <= n; i++)
            {
                ans = ans * (m - i + 1) / i;
            }

            return ans;
        }

        static void Main(string[] args)
        {
            int N = int.Parse(Console.ReadLine());
            string[] S = new string[N];
            int stlen = 0;
            for (int j = 0; j < N; j++)
            {
                S[j] = Console.ReadLine();
                stlen += S[j].Length;
            }
            var combinations = Combination.Enumerate(S, N, withRepetition: true);
            int i = 1;
            var ssary = Enumerable.Repeat<string>("(", stlen / 2).ToArray();
            var seary = Enumerable.Repeat<string>(")", stlen / 2).ToArray();
            var anss = string.Join("", ssary) + string.Join("", seary);
            foreach (var elem in combinations)
            {
                string s = string.Join("", elem.Select(x => x.ToString()).ToArray());
                if (anss == s)
                {
                    Console.WriteLine("Yes");
                    return;
                }
                // Console.WriteLine("{0} : {1}", i++, s);
            }

            Console.WriteLine("No");
        }
    }
    public static class Combination
    {
        public static IEnumerable<T[]> Enumerate<T>(IEnumerable<T> items, int k, bool withRepetition)
        {
            if (k == 1)
            {
                foreach (var item in items)
                    yield return new T[] { item };
                yield break;
            }
            foreach (var item in items)
            {
                var leftside = new T[] { item };

                // item よりも前のものを除く （順列と組み合わせの違い)
                // 重複を許さないので、unusedから item そのものも取り除く
                var unused = withRepetition ? items : items.SkipWhile(e => !e.Equals(item)).Skip(1).ToList();

                foreach (var rightside in Enumerate(unused, k - 1, withRepetition))
                {
                    yield return leftside.Concat(rightside).ToArray();
                }
            }
        }
    }
}