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 Rust (1.42.0) (judge verdict: Wrong Answer)

use proconio::input;
use proconio::marker::Chars;

fn main() {
    input! {
        n: usize,
        s: [Chars; n],
    };
    let mut count_all = (0, 0);
    for s_i in s.iter() {
        let mut count = (0, 0); // (open, close)
        let mut stack = vec![];
        for &c in s_i.iter() {
            match c {
                '(' => {
                    stack.push(c);
                    count.0 += 1;
                }
                ')' => {
                    if let Some('(') = stack.last() {
                        stack.pop();
                        count.0 -= 1;
                    } else {
                        stack.push(c);
                        count.1 += 1;
                    }
                }
                _ => unreachable!(),
            }
        }
        match count {
            (0, 0) => {
                // do nothing
            }
            (0, c) => {
                let (ao, ac) = count_all;
                count_all.0 = std::cmp::max(0, ao - c);
                count_all.1 = std::cmp::max(0, c - ao) + ac;
            }
            (o, 0) => {
                let (ao, ac) = count_all;
                count_all.1 = std::cmp::max(0, ac - o);
                count_all.0 = std::cmp::max(0, o - ac) + ao;
            }
            (o, c) => {
                let (ao, ac) = count_all;
                if ao + o >= ac + c {
                    count_all.0 = if ao >= c { ao - c } else { 0 } + o;
                    count_all.1 = std::cmp::max(0, c - ao) + ac;
                } else {
                    count_all.1 = if ac >= o { ac - o } else { 0 } + c;
                    count_all.0 = std::cmp::max(0, o - ac) + ao;
                }
            }
        }
    }
    let ans = if count_all == (0, 0) { "Yes" } else { "No" };
    println!("{}", ans);
}