A16051 | Bracket Coloring

A regular bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters "1" and "+" between the original characters of the sequence. For example:

- the bracket sequences "()()" and "(())" are regular (the resulting expressions are: "(1)+(1)" and "((1+1)+1)");
- the bracket sequences ")(", "(" and ")" are not.

A bracket sequence is called beautiful if one of the following conditions is satisfied:

- it is a regular bracket sequence;
- if the order of the characters in this sequence is reversed, it becomes a regular bracket sequence.

For example, the bracket sequences "()()", "(())", ")))(((", "))()((" are beautiful.

You are given a bracket sequence $s$ . You have to color it in such a way that:

- every bracket is colored into one color;
- for every color, there is at least one bracket colored into that color;
- for every color, if you write down the sequence of brackets having that color in the order they appear, you will get a beautiful bracket sequence.

Color the given bracket sequence $s$ into the minimum number of colors according to these constraints, or report that it is impossible.

The first line contains one integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases.

Each test case consists of two lines. The first line contains one integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ) — the number of characters in $s$ . The second line contains $s$ — a string of $n$ characters, where each character is either "(" or ")".

Additional constraint on the input: the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, print the answer as follows:

- if it is impossible to color the brackets according to the problem statement, print $-1$ ;
- otherwise, print two lines. In the first line, print one integer $k$ ( $1 \le k \le n$ ) — the minimum number of colors. In the second line, print $n$ integers $c_1, c_2, \dots, c_n$ ( $1 \le c_i \le k$ ), where $c_i$ is the color of the $i$ -th bracket. If there are multiple answers, print any of them.