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 PHP (7.4.4) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using ld = long double;
#define int ll
#define double ld
#define repd(i,a,b) for (int i=a;i<int(b);i++)
#define rep(i,n) repd(i,0,n)
#define VARNAME(v) #v
using P = pair<int,int>;
// constexpr  int MOD = 1000000007;
constexpr  int MOD = 998244353;
constexpr ll INF = 1e16;
#pragma region tmp
constexpr double EPS = 1e-10;
constexpr double PI = 3.141592653589793;
const string endn = "\n";
const string abc="abcdefghijklmnopqrstuvwxyz";
const string ABC="ABCDEFGHIJKLMNOPQRSTUVWXYZ";
template<class T1, class T2>ostream &operator<<(ostream &out, const pair<T1, T2> &o){out << "(" << o.first << ", " << o.second << ")";return out;}
template<class T> ostream &operator<<(ostream &out, const vector<T> &o){for(auto itr=o.begin();itr != o.end();itr++)out << *itr << (itr+1 != o.end() ? " " : "");return out;}
template<class T> ostream &operator<<(ostream &out, const deque<T> &o){for(auto itr=o.begin();itr != o.end();itr++)out << *itr << (itr+1 != o.end() ? " " : "");return out;}
template<class T> ostream &operator<<(ostream &out, const set<T> &o){out << "{";for(auto itr=o.begin();itr != o.end();itr++)out << *itr << (next(itr,1) != o.end() ? "," : "");out << "}";return out;}
template<class T> ostream &operator<<(ostream &out, const multiset<T> &o){out << "{";for(auto itr=o.begin();itr != o.end();itr++)out << *itr << (next(itr,1) != o.end() ? "," : "");out << "}";return out;}
template<class T, class U> ostream &operator<<(ostream &out, const map<T, U> &o){out << "{";for(auto itr=o.begin();itr != o.end();itr++)out << *itr << (next(itr,1) != o.end() ? " " : "");out << "}";return out;}
template<class T, class U> ostream &operator<<(ostream &out, const multimap<T, U> &o){out << "{";for(auto itr=o.begin();itr != o.end();itr++)out << *itr << (next(itr,1) != o.end() ? " " : "");out << "}";return out;}
template<class T> ostream &operator<<(ostream &out, queue<T> o){while(!o.empty())out << o.front() << (o.size() > 0 ? " " : ""),o.pop();return out;}
template<class T, class U> ostream &operator<<(ostream &out, priority_queue<T, vector<T>, U> o){while(!o.empty())out << o.top() << (o.size() > 0 ? " " : ""),o.pop();return out;}
template<class T> ostream &operator<<(ostream &out, stack<T> o){while(!o.empty())out << o.top() << (o.size() > 0 ? " " : ""),o.pop();return out;}
template<class T> inline bool chmin(T& a, T b){if(a>b){a = b;return 1;}return 0;}
template<class T> inline bool chmax(T& a, T b){if(a<b){a = b;return 1;}return 0;}
template<class T> inline bool chmin(T& a, initializer_list<T> b){T c = min(b);if(a>c){a = c;return 1;}return 0;}
template<class T> inline bool chmax(T& a, initializer_list<T> b){T c = max(b);if(a<c){a = c;return 1;}return 0;}
inline int mrep(int& a, int M=MOD){a%=M;return a=(a<0 ? a+M : a);}
inline int add(int& a, int b, int M=MOD){return a=(mrep(a,M)+mrep(b,M))%M;}
inline int mul(int& a, int b, int M=MOD){return a=(mrep(a,M)*mrep(b,M))%M;}
inline int add(int& a, initializer_list<int> b, int M=MOD){return a=(mrep(a,M)+accumulate(b.begin(), b.end(), (int)0, [&M](int acc, int i){return (acc+mrep(i,M))%M;}))%M;}
inline int mul(int& a, initializer_list<int> b, int M=MOD){return a=(mrep(a,M)*accumulate(b.begin(), b.end(), (int)1, [&M](int acc, int i){return (acc*mrep(i,M))%M;}))%M;}
inline int modpow(int b, int e, int M=MOD){int ret=1;while(e > 0){if(e%2)mul(ret,b,M);mul(b,b,M);e /= 2;};return ret;}
inline int modinv(int a, int M=MOD){int b=M,u=1,v=0;while(b){int t=a/b;a-=t*b;swap(a,b);u-=t*v;swap(u,v);}return mrep(u,M);}
template<class T> inline void print(T a){cout << a << endl;}
template<class T> inline void print(vector<T> a, int n){chmin(n,int(a.size()));rep(i,n)cout << a[i] << (i != n-1 ? " " : "\n");}
template<class T, size_t SIZE> inline void print(T (&a)[SIZE], int n=SIZE){chmin(n,int(SIZE));rep(i,n)cout << a[i] << (i != n-1 ? " ":"\n");}
inline void yesno(bool b){cout << (b ? "Yes" : "No") << endl;}
#pragma #endregion tmp

int n;
vector<P> seg;
vector<int> id;
bool fmax(P a, P b){
  return (a.first+a.second > b.first+b.second);
}

pair<P, int> getmax(int l, int r){
  P ret = P(-INF, -INF);int ind = -1;l += n;r += n;
  while(l < r){
    if(l&1){
      if(!fmax(ret, seg[l++])){
        ret = seg[l-1];
        ind = id[l-1];
      }
    };
    if(r&1){
      if(!fmax(ret, seg[--r])){
        ret = seg[r];
        ind = id[r];
      };
    }
    l >>= 1;r >>=1;
  }
  return make_pair(ret, ind);
}

void update(int i, P x){
  i += n;
  seg[i] = x;
  id[i] = i-n;
  while(i > 1){
    i >>= 1;
    if(fmax(seg[i<<1|0], seg[i<<1|1])){
      seg[i] = seg[i<<1|0];id[i] = id[i<<1|0];

    }else{
      seg[i] = seg[i<<1|1];id[i] = id[i<<1|1];
    }
  }
}
signed main(){
  cin>>n;P p[1100000];
  seg.resize(2*n);
  id.resize(2*n);
  rep(i,n){
    string s;cin>>s;
    int m = 0, sum = 0;
    rep(i,s.length()){
      if(s[i] == '('){
        sum++;
      }else{
        sum--;
      }
      chmin(m,sum);
    }
    p[i] = P(m, sum-m);
  }
  
  int sum = 0;bool b = true;
  sort(p,p+n,greater<P>());
  rep(i,n)update(i, p[i]);
  rep(_,n){
    int ok = -1, ng = n;
    while(abs(ok-ng) > 1){
      int mid = (ok+ng)/2;
      if(p[mid].first + sum >= 0)ok = mid;
      else ng = mid;
    }
    if(ok == -1){
      b = false;break;
    }
    auto mm = getmax(0, ok+1);P m = mm.first;int d = mm.second;sum += (m.first + m.second);
    update(d, P(m.first, -INF));
  }
  yesno(b && sum == 0);
}