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: Accepted)

#[allow(unused_imports)]
use proconio::{fastout, input, marker::Chars, marker::Usize1};

fn chmin<T>(a: &mut T, b: T)
where
    T: std::cmp::PartialOrd,
{
    if *a > b {
        *a = b;
    }
}

fn solve(s: Vec<Vec<char>>) -> Option<usize> {
    let mut buck = Vec::new();
    for s in s {
        let mut h = 0;
        let mut min = 0;
        for c in s {
            match c {
                '(' => {
                    h += 1;
                }
                ')' => {
                    h -= 1;
                }
                _ => unreachable!(),
            }
            chmin(&mut min, h);
        }
        let min = -min as usize;
        while buck.len() <= min {
            buck.push(Vec::new());
        }
        buck[min].push(h as usize);
    }
    let mut h = 0;
    for (i, b) in buck.into_iter().enumerate() {
        if i > h {
            return None;
        }
        for b in b {
            h += b;
        }
    }
    Some(h)
}

fn is_up(s: &Vec<char>) -> bool {
    s.iter().filter(|&&c| c == '(').count() >= s.iter().filter(|&&c| c == ')').count()
}

#[fastout]
fn main() {
    input! {
        n: usize,
        s: [Chars; n],
    }
    let up = s.iter().filter(|&s| is_up(s)).cloned().collect::<Vec<_>>();
    let down = s
        .iter()
        .filter(|&s| !is_up(s))
        .cloned()
        .map(|s| {
            s.into_iter()
                .rev()
                .map(|c| match c {
                    '(' => ')',
                    ')' => '(',
                    _ => unreachable!(),
                })
                .collect::<Vec<_>>()
        })
        .collect::<Vec<_>>();
    println!(
        "{}",
        match (solve(up), solve(down)) {
            (Some(u), Some(d)) if u == d => "Yes",
            _ => "No",
        }
    );
}