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)

#![allow(unused_imports)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), std::io::stderr());
            writeln!(err, "{} = {:?}", e, $e).unwrap()
        })*
    };
}

fn main() {
    let n = read::<usize>();
    let mut ss = vec![];
    for i in 0..n {
        let s = read::<String>();
        ss.push(s);
    }

    let mut pt_indexes = vec![];
    let mut nt_indexes = vec![];
    for (i, ref s) in ss.iter().enumerate() {
        let mut cur = 0i64;
        for ch in s.chars() {
            if ch == '(' {
                cur += 1;
            } else {
                cur -= 1;
            }
        }
        if cur >= 0 {
            pt_indexes.push(i)
        } else {
            nt_indexes.push(i);
        }
    }

    let mut pairs = vec![];
    for i in pt_indexes {
        let s = &ss[i];
        let mut cur = 0i64;
        let mut worst = 0i64;
        for ch in s.chars() {
            if ch == '(' {
                cur += 1;
            } else {
                cur -= 1;
            }
            worst = min(worst, cur);
        }
        pairs.push((worst, cur));
    }
    pairs.sort_by_key(|x| -x.0);

    let mut cur_level1 = 0;
    for (worst, total) in pairs {
        if cur_level1 < -worst {
            println!("No");
            return;
        }
        cur_level1 += total;
    }

    let mut pairs = vec![];
    for i in nt_indexes {
        let s = &ss[i];
        let mut cur = 0i64;
        let mut worst = 0i64;
        for ch in s.chars() {
            if ch == ')' {
                cur += 1;
            } else {
                cur -= 1;
            }
            worst = min(worst, cur);
        }
        pairs.push((worst, cur));
    }
    pairs.sort_by_key(|x| -x.0);

    let mut cur_level2 = 0;
    for (worst, total) in pairs {
        if cur_level2 < -worst {
            println!("No");
            return;
        }
        cur_level2 += total;
    }

    if cur_level1 == cur_level2 && cur_level1 >= 0 {
        println!("Yes");
    } else {
        println!("No");
    }
}

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}