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++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;
#define MOD (long long int)(1e9+7)
#define ll long long int
#define rep(i,n) for(int i=0; i<(int)(n); i++)
#define reps(i,n) for(int i=1; i<=(int)(n); i++)
#define REP(i,n) for(int i=n-1; i>=0; i--)
#define REPS(i,n) for(int i=n; i>0; i--)
#define FOR(i,a,b) for(int i=a; i<(int)(b); i++)
#define ALL(x) (x).begin(),(x).end()
#define RALL(a) (a).rbegin(), (a).rend()
#define CLR(a) memset((a), 0 ,sizeof(a))
#define PB push_back
#define MP make_pair
#define SP << " " <<
const int INF = 1001001001;
const ll LINF = 100100100100100100;
const double EPS = 1e-10;
const long double PI  = acos(-1.0L);
typedef pair<int,int> PII;
typedef vector<int> VI;
typedef vector<VI> VVI;
typedef vector<ll> VL;
#define chmax(a,b) a = (((a)<(b))?(b):(a))
#define chmin(a,b) a = (((a)>(b))?(b):(a))

__attribute__((constructor))
void initial(){
	cin.tie(nullptr);
	ios::sync_with_stdio(false);
	cout << fixed << setprecision(15);
}

signed main(){
	int n; cin>>n;
	int l=0,r=0;
	vector<string> s(n);
	bool start=false,end=false;
	bool mid=true;
	rep(i,n){
		cin>>s[i];
	}
	rep(i,n){
		int tmpl=0,tmpr=0;
		rep(j,s[i].size()){
			if(s[i][j]=='(') tmpl++;
			else tmpr++;
		}
		if(tmpl>tmpr&&s[i][0]=='(') start=true;
		if(tmpl<tmpr&&s[i][s[i].size()-1]==')') end=true;
		if(s[i][0]!='('||s[i][s[i].size()-1]!=')') mid=false;
		l+=tmpl; r+=tmpr;
	}
	if(l==r){
		if(mid||(start&&end)) return cout<<"Yes"<<endl,0;
	}
	cout<<"No"<<endl;
	return 0;
}