F - Bracket Sequencing

Score : $600$ points

Problem Statement

A bracket sequence is a string that is one of the following:

1. An empty string;
2. The concatenation of (, $A$, and ) in this order, for some bracket sequence $A$ ;
3. 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.

2
)
(()

Yes

Concatenating (() and ) in this order forms a bracket sequence.

2
)(
()

No

4
((()))
((((((
))))))
()()()

Yes

3
(((
)
)

No