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 Java (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.*;
import java.util.*;
import java.math.*;
import java.awt.Point;
 
public class Main {
	static final long MOD = 998244353L;
	//static final long INF = 1000000000000000007L;
	//static final long MOD = 1000000007L;
	static final int INF = 1000000007;
	
	static long[] fac;
	static long[] invFac;
	public static void main(String[] args) {
		FastScanner sc = new FastScanner();
		PrintWriter pw = new PrintWriter(System.out);
		
		int N = sc.ni();
		int totPref = 0;
		
		PriorityQueue<int[]> pos = new PriorityQueue<int[]>(new Comparator<int[]>(){
			public int compare(int[] a, int[] b) {
				return b[1]-a[1];
			}
		});
		PriorityQueue<int[]> neg = new PriorityQueue<int[]>(new Comparator<int[]>(){
			public int compare(int[] a, int[] b) {
				int nA = a[0]-a[1];
				int nB = b[0]-b[1];
				return (nB-nA);
			}
		});
		for (int i = 0; i < N; i++) {
			String s = sc.next();
			int p = 0;
			int m = 0;
			for (char c: s.toCharArray()) {
				if (c == '(') {
					p++;
				} else {
					p--;
					m = Math.min(m,p);
				}
			}
			int[] pair = new int[]{p,m};
			if (p >= 0)
				pos.add(pair);
			else
				neg.add(pair);
			totPref += p;
		}
		if (totPref != 0) {
			pw.println("No");
			pw.close();
			return;
		}
		
		int pref = 0;
		
		while (!pos.isEmpty()) {
			int[] pair = pos.poll();
			if (pref+pair[1] >= 0) {
				//valid
				pref += pair[0];
			} else {
				pw.println("No");
				pw.close();
				return;
			}
		}
		while (!neg.isEmpty()) {
			int[] pair = neg.poll();
			//pw.println(Arrays.toString(pair));
			if (pref+pair[1] >= 0) {
				//valid
				pref += pair[0];
			} else {
				pw.println("No");
				pw.close();
				return;
			}
		}
		
		pw.println("Yes");
		
		pw.close();
	}
 
	public static long dist(long[] p1, long[] p2) {
		return (Math.abs(p2[0]-p1[0])+Math.abs(p2[1]-p1[1]));
	}
	
	//Find the GCD of two numbers
	public static long gcd(long a, long b) {
		if (a < b) return gcd(b,a);
		if (b == 0)
			return a;
		else
			return gcd(b,a%b);
	}
	
	//Fast exponentiation (x^y mod m)
	public static long power(long x, long y, long m) { 
		if (y < 0) return 0L;
		long ans = 1;
		x %= m;
		while (y > 0) { 
			if(y % 2 == 1) 
				ans = (ans * x) % m; 
			y /= 2;  
			x = (x * x) % m;
		} 
		return ans; 
	}
	
	public static int[] shuffle(int[] array) {
		Random rgen = new Random();
		for (int i = 0; i < array.length; i++) {
		    int randomPosition = rgen.nextInt(array.length);
		    int temp = array[i];
		    array[i] = array[randomPosition];
		    array[randomPosition] = temp;
		}
		return array;
	}
	
	public static long[] shuffle(long[] array) {
		Random rgen = new Random();
		for (int i = 0; i < array.length; i++) {
		    int randomPosition = rgen.nextInt(array.length);
		    long temp = array[i];
		    array[i] = array[randomPosition];
		    array[randomPosition] = temp;
		}
		return array;
	}
	
	public static int[][] shuffle(int[][] array) {
		Random rgen = new Random();
		for (int i = 0; i < array.length; i++) {
		    int randomPosition = rgen.nextInt(array.length);
		    int[] temp = array[i];
		    array[i] = array[randomPosition];
		    array[randomPosition] = temp;
		}
		return array;
	}
	
    public static int[][] sort(int[][] array) {
    	//Sort an array (immune to quicksort TLE)
 
		Arrays.sort(array, new Comparator<int[]>() {
			  @Override
        	  public int compare(int[] a, int[] b) {
				  return a[0]-b[0]; //ascending order
	          }
		});
		return array;
	}
    
    public static long[][] sort(long[][] array) {
    	//Sort an array (immune to quicksort TLE)
		Random rgen = new Random();
		for (int i = 0; i < array.length; i++) {
		    int randomPosition = rgen.nextInt(array.length);
		    long[] temp = array[i];
		    array[i] = array[randomPosition];
		    array[randomPosition] = temp;
		}
		Arrays.sort(array, new Comparator<long[]>() {
			  @Override
        	  public int compare(long[] a, long[] b) {
				  if (a[0] < b[0])
					  return -1;
				  else if (a[0] > b[0])
					  return 1;
				  else
					  return 0;
	          }
		});
		return array;
	}
    
    static class FastScanner { 
        BufferedReader br; 
        StringTokenizer st; 
  
        public FastScanner() { 
            br = new BufferedReader(new InputStreamReader(System.in)); 
        } 
  
        String next() { 
            while (st == null || !st.hasMoreElements()) { 
                try { 
                    st = new StringTokenizer(br.readLine());
                } catch (IOException  e) { 
                    e.printStackTrace(); 
                } 
            } 
            return st.nextToken(); 
        }
        
        int ni() { 
            return Integer.parseInt(next()); 
        }
  
        long nl() { 
            return Long.parseLong(next()); 
        } 
  
        double nd() { 
            return Double.parseDouble(next()); 
        } 
  
        String nextLine() {
            String str = ""; 
            try { 
                str = br.readLine(); 
            } catch (IOException e) {
                e.printStackTrace(); 
            } 
            return str;
        }
    }
}