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: Wrong Answer)

using System;
using System.IO;
using System.Linq;

namespace AtCoder.R167.F
{
    public static class Solver
    {
        public static void Main()
        {
            using (var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false })
            {
                Solve(Console.In, sw);
            }
        }

        public static void Solve(TextReader tr, TextWriter tw)
        {
            WriteAnswer(tw, Parse(tr));
        }

        private static void WriteAnswer(TextWriter tw, string answer)
        {
            tw.WriteLine(answer);
        }

        private static string Parse(TextReader tr)
        {
            var n = int.Parse(tr.ReadLine());

            var s = new string[n];
            for (int i = 0; i < n; ++i)
            {
                s[i] = tr.ReadLine();
            }
            
            return Calc(s);
        }

        public static string Calc(string[] s)
        {
            var t = new (int, int)[s.Length];

            for (int i = 0; i < s.Length; ++i)
            {
                foreach (var c in s[i])
                {
                    if (c == '(')
                    {
                        ++t[i].Item2;
                    }
                    else if (c == ')')
                    {
                        t[i].Item1 = Math.Min(t[i].Item1, --t[i].Item2);
                    }
                }
            }

            if (t.Sum(x => x.Item2) != 0)
                return "No";

            var upper = t.Where(x => x.Item2 >= 0).OrderByDescending(x => x.Item1).ToArray();

            var currentLevel = 0;

            foreach (var u in upper)
            {
                if (u.Item1 + currentLevel < 0)
                {
                    return "No";
                }
                currentLevel += u.Item2;
            }

            return "Yes";
        }
    }
}