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>
#define For(i, a, b) for(int (i)=(int)(a); (i)<(int)(b); ++(i))
#define rFor(i, a, b) for(int (i)=(int)(a)-1; (i)>=(int)(b); --(i))
#define rep(i, n) For((i), 0, (n))
#define rrep(i, n) rFor((i), (n), 0)
#define fi first
#define se second
#define double long double
using namespace std;
typedef long long lint;
typedef unsigned long long ulint;
typedef pair<int, int> pii;
typedef pair<lint, lint> pll;
template<class T> bool chmax(T &a, const T &b){if(a<b){a=b; return true;} return false;}
template<class T> bool chmin(T &a, const T &b){if(a>b){a=b; return true;} return false;}
template<class T> T div_floor(T a, T b){
    if(b < 0) a *= -1, b *= -1;
    return a>=0 ? a/b : (a+1)/b-1;
}
template<class T> T div_ceil(T a, T b){
    if(b < 0) a *= -1, b *= -1;
    return a>0 ? (a-1)/b+1 : a/b;
}

constexpr lint mod = 1e9+7;
constexpr lint INF = mod * mod;
constexpr int MAX = 200010;

template<typename T> struct SegTree{
    using F=function<T(T, T)>;
    int sz=1, seq_sz;
    T et;
    F f, g;
    vector<T> node;

    SegTree(int sz_, T et_, F f_, F g_): seq_sz(sz_), et(et_), f(f_), g(g_){
        while(sz<sz_) sz<<=1;
        node.resize(sz<<1, et);
    }

    void build(vector<T> &a){
        rep(i, a.size()) node[i+sz] = a[i];
        rFor(i, sz, 1) node[i] = f(node[i<<1], node[(i<<1)+1]);
    }

    void build(T x){
        rep(i, seq_sz) node[i+sz] = x;
        rFor(i, sz, 1) node[i] = f(node[i<<1], node[(i<<1)+1]);
    }

    void update(int i, T x){
        i+=sz;
        node[i]=g(node[i], x);
        i>>=1;
        while(i){
            node[i]=f(node[i<<1], node[(i<<1)+1]);
            i>>=1;
        }
    }

    T query(int l, int r){
        T vl=et, vr=et;
        for(l+=sz, r+=sz; l<r; l>>=1, r>>=1){
            if(l&1) vl=f(vl, node[l++]);
            if(r&1) vr=f(node[--r], vr);
        }
        return f(vl, vr);
    }

    int search_left(int l, const T val, function<bool(T, T)> cmp, function<bool(T, T)> check){
        T sum = et;
        for(l+=sz; ; l>>=1){
            if(check(f(sum, node[l]), val)){
                while(l < sz){
                    if(check(f(sum, node[l<<1]), val)) l<<=1;
                    else{
                        sum = f(sum, node[l<<1]);
                        l = (l<<1) + 1;
                    }
                }
                return l-sz;
            }
            if(__builtin_popcount(l+1) == 1) return seq_sz;
            if(l&1) sum = f(sum, node[l++]);
        }
    }

    int search_right(int r, const T val, function<bool(T, T)> cmp, function<bool(T, T)> check){
        T sum = et;
        for(r+=sz; ; r>>=1){
            if(check(f(sum, node[r]), val)){
                while(r < sz){
                    if(check(f(node[(r<<1)+1], sum), val)) r = (r<<1) + 1;
                    else{
                        sum = f(node[(r<<1)+1], sum);
                        r<<=1;
                    }
                }
                return r-sz;
            }
            if(__builtin_popcount(r) == 1) return -1;
            if(!(r&1)) sum = f(node[r--], sum);
        }
    }

    int lower_bound_left(int l, const T val, function<bool(T, T)> cmp=less<>()){
        auto check = [&cmp](auto a, auto b){return !cmp(a, b);};
        return search_left(l, val, cmp, check);
    }

    int upper_bound_left(int l, const T val, function<bool(T, T)> cmp=less<>()){
        auto check = [&cmp](auto a, auto b){return cmp(b, a);};
        return search_left(l, val, cmp, check);
    }

    int lower_bound_right(int l, const T val, function<bool(T, T)> cmp=greater<>()){
        auto check = [&cmp](auto a, auto b){return !cmp(b, a);};
        return search_right(l, val, cmp, check);
    }

    int upper_bound_right(int l, const T val, function<bool(T, T)> cmp=greater<>()){
        auto check = [&cmp](auto a, auto b){return cmp(a, b);};
        return search_right(l, val, cmp, check);
    }
};

pii sp(string &s){
    int sum = 0, m = 0;
    rep(i, s.size()){
        if(s[i] == '(') ++sum;
        else --sum;
        chmin(m, sum);
    }
    return {m, sum};
}

int main(){
    int n;
    cin >> n;
    pii p[n];
    rep(i, n){
        string s;
        cin >> s;
        p[i] = sp(s);
    }
    sort(p, p+n);

    auto f = [](pii a, pii b){
        if(a.fi == b.fi) return a.se < b.se ? a : b;
        return max(a, b);
    };
    auto g = [](pii a, pii b){return b;};
    SegTree<pii> st(n, {-mod, mod}, f, g);
    rep(i, n) st.update(i, {p[i].se, i});

    int tsum = 0;
    rep(_, n){
        int L = lower_bound(p, p+n, make_pair(-tsum, -(int)mod)) - p;
        if(L == n){
            puts("No");
            return 0;
        }
        auto t = st.query(L, n);
        if(t.se == mod){
            puts("No");
            return 0;
        }
        tsum += t.fi;
        st.update(t.se, {-mod, mod});
    }
    puts(tsum == 0 ? "Yes" : "No");
}