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constructive algorithms

data structures

greedy

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D. Bracket Coloring

time limit per test

2 secondsmemory limit per test

512 megabytesinput

standard inputoutput

standard outputA 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.

Input

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 $$$2 \cdot 10^5$$$.

Output

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.

Example

Input

48((())))(4(())4))((3(()

Output

2 2 2 2 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 -1

Codeforces (c) Copyright 2010-2023 Mike Mirzayanov

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