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.Linq;
using System.Collections;
using System.Collections.Generic;

public class AtCoder
{
  public static void Main()
  {
    Func<List<(int, int)>, bool> check = (s) =>
    {
      int h = 0;
      foreach (var p in s)
      {
        int b = h + p.Item1;
        if (b < 0) {
          return false;
        }
        h += p.Item2;
      }
      return true;
    };

    var n = int.Parse(Console.ReadLine().Trim());

    var ls = new List<(int, int)>();
    var rs = new List<(int, int)>();
    int total = 0;
    for (var i = 0; i < n; i++)
    {
      var s = Console.ReadLine().Trim();

      int h = 0, b = 0;
      foreach (var c in s)
      {
        if (c == '(')
        {
          ++h;
        }
        else
        {
          --h;
        }
        b = Math.Min(b, h);
      }
      if (h > 0)
      {
        ls.Add((b, h));
      }
      else { 
        rs.Add((b - h, -h)); 
      }
      total += h;
    }

    ls.OrderByDescending(e => e.Item1);
    rs.OrderByDescending(e => e.Item1);

    if (check(ls) && check(rs) && (total == 0))
    {
      Console.WriteLine("Yes");
    }
    else
    {
      Console.WriteLine("No");
    }
  }
}