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 PHP (7.4.4) (judge verdict: Runtime Error)

<?php
$N = (int)fgets(STDIN);
$S = [];
$prefix = false;
$suffix = false;
$num_of_L = 0;
$num_of_R = 0;

for ($i = 0; $i < $N; $i++) {
    $s = removeBracketRow(trim(fgets(STDIN)));
    $num_of_L += mb_substr_count($s, '(');
    $num_of_R += mb_substr_count($s, ')');
    if ($s[0] === '(') {
        $prefix = true;
    }
    if ($s[strlen($s)-1] === ')') {
        $suffix = true;
    }
}

if (($num_of_L === 0 && $num_of_R === 0) || ($prefix && $suffix && $num_of_L === $num_of_R)) {
    echo 'Yes';
    return;
}
echo 'No';


function removeBracketRow($s) {
    $old_s = '';
    while ($s !== $old_s) {
        $old_s = $s;
        $s = str_replace('()', '', $s);
    }
    return $s;
}