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)

// #pragma GCC target("avx")  // CPU 処理並列化
// #pragma GCC optimize("O3")  // CPU 処理並列化
// #pragma GCC optimize("unroll-loops")  // 条件処理の呼び出しを減らす
#include <bits/stdc++.h>
#include <complex>

// #include <iostream> // cout, endl, cin
// #include <string> // string, to_string, stoi
// #include <vector> // vector
// #include <algorithm> // min, max, swap, sort, reverse, lower_bound, upper_bound
// #include <utility> // pair, make_pair
// #include <tuple> // tuple, make_tuple
// #include <cstdint> // int64_t, int*_t
// #include <cstdio> // printf
// #include <map> // map
// #include <queue> // queue, priority_queue
// #include <set> // set
// #include <stack> // stack
// #include <deque> // deque
// #include <unordered_map> // unordered_map
// #include <unordered_set> // unordered_set
// #include <bitset> // bitset
// #include <climits>
// #include <cmath>
// #include <iomanip>
// #include <functional>
// #include <numeric>
// #include <random>
 
using namespace std;

#define int long long
#define pb push_back
#define eb emplace_back
// #define F first
// #define S second
#define FOR(i,a,b) for(int (i)=(a);(i)<(int)(b);(i)++)
#define rep(i,n) FOR(i,0,n)
#define RFOR(i,a,b) for(int (i)=(a);(i)>=(int)(b);(i)--)
#define rrep(i,n) RFOR(i,n,0)
#define all(a) (a).begin(),(a).end()
#define rall(a) (a).rbegin(),(a).rend()
#define ve vector
#define vi vector<int>
#define vp vector<pair<int,int>>
#define vvi vector<vector<int>>
#define UNIQUE(a) sort(all(a)), a.erase(unique(all(a)), a.end())
#define Double double
// #define endl '\n'
 
template<typename T> using pq = priority_queue<T,vector<T>,greater<T>>; 
using ll = long long;
using UnWeightedGraph = vector<vector<int>>;
ll INF = LLONG_MAX / 4 - 100;
int IINF = INT_MAX / 4;
ll mod = 1e9 + 7;
int dx[] = {1,0,-1,0}, dy[] = {0,1,0,-1};
vector<ll> prime;
 
long double pi = 3.141592653589793238;
    
class fact {
public:
    long long fmod = 1e9+7;
    vector<long long> fac, finv, inv;
    fact (int n, long long Mod = 1e9+7) {
        fmod = Mod;
        fac = vector<long long>(n + 1, 0);
        finv = vector<long long>(n + 1, 0);
        inv = vector<long long>(n + 1, 0);
        fac[0] = fac[1] = 1;
        finv[0] = finv[1] = 1;
        inv[1] = 1; 
        for (int i = 2; i < n + 1; i++) {
            fac[i] = fac[i-1] * i % fmod;
            inv[i] = mod - inv[mod%i] * (mod/i) % mod;
            finv[i] = finv[i-1] * inv[i] % mod;
        }
    }
    ll nCr(ll n, ll r) {if(n < r) return 0; return fac[n] * finv[r] % fmod * finv[n-r] % fmod;}
    ll POW(ll a, ll b) {ll c = 1; while (b > 0) {if (b & 1) {c = a * c%fmod;}a = a * a%fmod; b >>= 1;}return c;}
    inline int operator [] (int i) {return fac[i];}
    ll DeBuG(ll n, ll r);
};
    
void DEBUG(vector<int> a) {for(int i=0;i<a.size();i++)cout<<a[i]<<" ";cout<<endl;}
void EMP(int x) {cout<<"!!!"<<x<<"!!!"<<endl;}
ll GCD(ll a, ll b) {ll c; while (b != 0) {c = a % b; a = b; b = c;}return a;}
ll LCM(ll a, ll b) {return (a / GCD(a, b)) * (b / GCD(a, b)) * GCD(a, b);}
ll POW(ll a, ll b) {ll c = 1; while (b > 0) {if (b & 1) {c = a * c%mod;}a = a * a%mod; b >>= 1;}return c;}
void PRI(ll n) {bool a[n + 1]; for (int i = 0; i < n + 1; i++) {a[i] = 1;}for (int i = 2; i < n + 1; i++) {if (a[i]) {prime.pb(i); ll b = i; while (b <= n) {a[b] = 0; b += i;}}}}
template <typename T> T chmin(T& a, T b) {if(a>b)a=b;return a;}
template <typename T> T chmax(T& a, T b) {if(a<b)a=b;return b;}
bool isSqrt(ll a) {return pow(sqrt(a),2) == a ? 1 : 0;}
void YesNo(bool a) {if (a) cout << "Yes"; else cout << "No"; cout << endl;}
void yesno(bool a) {if (a) cout << "yes"; else cout << "no"; cout << endl;}
void YESNO(bool a) {if (a) cout << "YES"; else cout << "NO"; cout << endl;}
double dis(int x1, int x2, int y1, int y2) {
    return sqrt((double)abs(x1-x2)*(double)abs(x1-x2)+(double)abs(y1-y2)*(double)abs(y1-y2));
}
int ceili(int x, int y) {
    if (x % y == 0) return x / y;
    else return x / y + 1;
}
unsigned int randxor() {
    static unsigned int x=123456789,y=362436069,z=521288629,w=88675123;
    unsigned int t;
    t=(x^(x<<11));x=y;y=z;z=w; return( w=(w^(w>>19))^(t^(t>>8)) );
}
 
uint64_t xor64(void) {
    static uint64_t x = 88172645463325252ULL;
    x = x ^ (x << 7);
    return x = x ^ (x >> 9);
}

template<typename Monoid>
class SegTree {

//半開区間!!!!!!!!!!!!!!!
public:
    int p = 2;
    vector<Monoid> node;
    using Func = function<Monoid(Monoid,Monoid)>;
    Func pick; Monoid init;//操作と単位元

    SegTree(vector<Monoid> A, string type, Monoid m) {
        if (type == "min") {
            pick = [](Monoid a, Monoid b){return min(a,b);};
        } else if (type == "max") {
            pick = [](Monoid a, Monoid b){return max(a,b);};
        } else if (type == "sum") {
            // pick = [](Monoid a, Monoid b){return a + b;};
        }

        init = m;
        while (A.size() > p) p *= 2;
        node = vector<Monoid>(2*p-1,init);

        for (int i = 0;i < A.size();i++) {
            node[p-1+i] = A[i];
        }

        for (int i = p - 2;i >= 0;i--) {
            node[i] = pick(node[i*2+1],node[i*2+2]);
        }
    }

    SegTree(vector<Monoid> A, Func f, Monoid m) {
        pick = f; init = m;
        while (A.size() > p) p *= 2;
        node = vector<Monoid>(2*p-1,init);

        for (int i = 0;i < A.size();i++) {
            node[p-1+i] = A[i];
        }

        for (int i = p - 2;i >= 0;i--) {
            node[i] = pick(node[i*2+1],node[i*2+2]);
        }
    }

    //1点更新
    void update(int i, Monoid v) {
        for (node[(i+=p)-1] = v; i >>= 1;) {
            node[i-1] = pick(node[i*2-1], node[i*2]);
        }
    }

    //1点加算
    void add(int i, Monoid v) {
        update(i, this[i] + v);
    }
 
    Monoid getval(int start,int end,int k = 0,int l = 0,int r = -1) {
        if (r < 0) r = p;
        if (r <= start || end <= l) {return init;}
        if (start <= l && r <= end) {return node[k];}
        Monoid L,R;
        L = getval(start,end,k*2+1,l,(l+r)/2);
        R = getval(start,end,k*2+2,(l+r)/2,r);
        return {pick(L,R)};
    }

    inline Monoid operator [] (int i) {
        return node[i+p-1];
    }
};

void solve() {
    int n; cin >> n;
    vector<pair<int, int>> ue, si;
    vector<string> s(n); rep (i, n) cin >> s[i];

    vector<vector<int>> A(n);
    int sum = 0;
    rep (i, n) {
        int now = 0, mi = 0;
        for (auto elm : s[i]) {
            if (elm == '(') now++;
            else now--, chmin(mi, now);
        }
        A[i] = {-mi, now, i};
        sum += now;
    }
    if (sum != 0) {
        cout << "No" << endl;
        return;
    }
    sort(all(A));
    vector<int> Mis(n); rep (i, n) Mis[i] = A[i][0];
    vector<vector<int>> His(n); rep (i, n) His[i] = {A[i][1], i};
    SegTree<vector<int>> seg(His, "max", {-INF, -INF});

    sum = 0;
    rep (i, n) {
        int ind = upper_bound(all(Mis), sum)-Mis.begin();
        auto res = seg.getval(0, ind);
        if (res[0] == -INF) {
            cout << "No" << endl;
            return;
        }
        sum += res[0];
        seg.update(res[1], {-INF, res[1]});
    }
    cout << "Yes" << endl;
}
 
signed main() {
    cin.tie(0);
    ios::sync_with_stdio(false);
    solve();
}