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 PyPy3 (7.3.0) (judge verdict: Time Limit Exceeded)

import sqlite3

conn = sqlite3.connect(":memory:", isolation_level=None, check_same_thread=False)
conn.executescript(
    "CREATE TABLE t (max_o INTEGER, max_c INTEGER); CREATE INDEX ind1 ON t(max_o); CREATE INDEX ind2 ON t(max_c);",
)

N = int(input())
Ss = []
for _ in range(N):
    S = input()
    max_c = 0
    tot_o = 0
    tot_c = 0
    for c in S:
        if c == "(":
            tot_o += 1
        else:
            tot_c += 1
            max_c = max(max_c, tot_c - tot_o)
    Ss.append((tot_o - tot_c + max_c, max_c))
conn.executemany("INSERT INTO t VALUES (?, ?)", Ss)
d = 0
for max_o, max_c in conn.execute("SELECT * FROM t WHERE max_o >= max_c ORDER BY max_c"):
    if max_c > d:
        d = -1
        break
    d += max_o - max_c
for max_o, max_c in conn.execute(
    "SELECT * FROM t WHERE max_o < max_c ORDER BY max_o DESC"
):
    if max_c > d:
        d = -1
        break
    d += max_o - max_c
if d == 0:
    print("Yes")
else:
    print("No")