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.

正则括号序列是指：通过在原序列的字符之间插入字符 "1" 和 "+"，可以将其转化为一个合法的算术表达式。例如：

• 括号序列 "()()" 和 "(())" 是正则的（对应的表达式分别为 "(1)+(1)" 和 "((1+1)+1)"）；
• 括号序列 ")(", "(" 和 ")" 不是正则的。

若一个括号序列满足以下任一条件，则称其为优美括号序列：

• 它本身是一个正则括号序列；
• 将该序列中字符的顺序反转后，得到的是一个正则括号序列。

例如，括号序列 "()()"、"(())"、")))((("、"))()(("` 都是优美的。

现给定一个括号序列 $s$。你需要对其进行染色，使得满足以下条件：

• 每个括号恰好被染成一种颜色；
• 每种使用的颜色至少染了一个括号；
• 对于每种颜色，若将该颜色的所有括号按其在原序列中出现的顺序写出，则所得括号序列是优美的。

请在满足上述约束的前提下，用最少的颜色数对给定括号序列 $s$ 进行染色；若不可能，请报告无解。

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 $2 \cdot 10^5$.

第一行包含一个整数 $t$（$1 \le t \le 10^4$）—— 测试用例的数量。

每个测试用例由两行组成。第一行包含一个整数 $n$（$2 \le n \le 2 \cdot 10^5$）—— 字符串 $s$ 的长度。第二行包含字符串 $s$，它由 $n$ 个字符组成，每个字符为 "(" 或 ")"。

输入的额外约束：所有测试用例的 $n$ 值之和不超过 $2 \cdot 10^5$。

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.

对于每个测试用例，按如下方式输出答案：

• 如果无法按照题目要求对括号进行染色，则输出 $-1$；
• 否则，输出两行。第一行输出一个整数 $k$（$1 \le k \le n$）—— 所需的最少颜色数；第二行输出 $n$ 个整数 $c_1, c_2, \dots, c_n$（$1 \le c_i \le k$），其中 $c_i$ 表示第 $i$ 个括号的颜色。若存在多个合法答案，输出任意一个即可。