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)]
use itertools::Itertools;
#[allow(unused_imports)]
use num::Integer;
#[allow(unused_imports)]
use proconio::derive_readable;
use proconio::fastout;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::cmp;

fn solve(ss: &Vec<(i32, i32)>) -> bool {
  let sum: i32 = ss.iter().map(|&(_, l)| l).sum();
  if sum != 0 {
    return false;
  }
  let mut gain: i32 = ss
    .iter()
    .filter_map(|&(b, l)| if b >= 0 { Some(l) } else { None })
    .sum();
  let (mut up, mut down): (Vec<(i32, i32)>, Vec<(i32, i32)>) = ss
    .iter()
    .filter(|&&(b, _)| b < 0)
    .partition(|&&(_, l)| l >= 0);
  up.sort_by_key(|&(b, _)| -b);
  down.sort_by_key(|&(b, l)| (b, -l));
  for &(b, l) in &up {
    if gain + b < 0 {
      return false;
    }
    gain += l;
  }
  for &(b, l) in &down {
    if gain + b < 0 {
      return false;
    }
    gain += l;
  }
  gain == 0
}

#[fastout]
fn main() {
  input! {
    n: usize,
    ss: [Chars; n],
  }
  let ss: Vec<_> = ss
    .into_iter()
    .map(|s| {
      let mut current = 0;
      let mut bottom = 0;
      for c in s.into_iter() {
        if c == '(' {
          current += 1;
        } else {
          current -= 1;
        }
        bottom = cmp::min(bottom, current);
      }
      (bottom, current)
    })
    .collect();

  println!("{}", if solve(&ss) { "Yes" } else { "No" });
}