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 std::io::*;
use std::str::FromStr;
use std::cmp::{ Ordering };

fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("faild to read char") as char)
        .take_while(|c| !c.is_whitespace())
        .collect();
    token
        .parse()
        .ok()
        .expect("faild to parse token")
}

fn main(){
    let n: usize = read();
    let mut s: Vec<Paren> = (0..n).map(|_| Paren::new(read::<String>())).collect();
    s.sort();

    let mut now: i64 = 0;

    for p in &s {
        if now <= 0 && p.end > 0 {
            println!("No");
            return;
        }
        now += p.start - p.end;
        if now < 0 {
            println!("No");
            return;
        }
    }
    if now == 0 {
        println!("Yes");
    } else {
        println!("No");
    }

}

#[derive(Clone, Debug, PartialEq, Eq)]
struct Paren {
    start: i64,
    end: i64,
}

impl Paren {
    fn new(s: String) -> Paren {
        let mut p = Paren { start: 0, end: 0, };
        for c in &s.chars().collect::<Vec<char>>() {
            if *c == '(' {
                p.start += 1;
            } else if p.start > 0 {
                p.start -= 1;
            } else {
                p.end += 1;
            }
        }
        p
    }
}

impl PartialOrd for Paren {
    fn partial_cmp(&self, other: &Paren) -> Option<Ordering> {
        if self.end == 0 && other.end == 0 {
            self.start.partial_cmp(&other.start)
        } else if self.end == 0 {
            Some(Ordering::Less)
        } else if other.end == 0 {
            Some(Ordering::Greater)
        } else if other.start == 0 && self.start == 0 {
            self.start.partial_cmp(&other.end)
        } else if self.start == 0 {
            Some(Ordering::Greater)
        } else if other.start == 0 {
            Some(Ordering::Less)
        } else {
            (self.end - self.start).partial_cmp(&(other.end - other.start))
        }
    }
}

impl Ord for Paren {
    fn cmp(&self, other: &Paren) -> Ordering {
        self.partial_cmp(other).unwrap()
    }
}