CF1837D 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.