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:

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

You are given two strings $$$s$$$ and $$$a$$$, the string $$$s$$$ has length $$$n$$$, the string $$$a$$$ has length $$$n - 3$$$. The string $$$s$$$ is a bracket sequence (i. e. each element of this string is either an opening bracket character or a closing bracket character). The string $$$a$$$ is a binary string (i. e. each element of this string is either 1 or 0).

The string $$$a$$$ imposes some constraints on the string $$$s$$$: for every $$$i$$$ such that $$$a_i$$$ is 1, the string $$$s_i s_{i+1} s_{i+2} s_{i+3}$$$ should be a regular bracket sequence. Characters of $$$a$$$ equal to 0 don't impose any constraints.

Initially, the string $$$s$$$ may or may not meet these constraints. You can perform the following operation any number of times: replace some character of $$$s$$$ with its inverse (i. e. you can replace an opening bracket with a closing bracket, or vice versa).

Determine if it is possible to change some characters in $$$s$$$ so that it meets all of the constraints, and if it is possible, calculate the minimum number of characters to be changed.

Input

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

Each test case consists of three lines. The first line contains one integer $$$n$$$ ($$$4 \le n \le 2 \cdot 10^5$$$). The second line contains the string $$$s$$$, consisting of exactly $$$n$$$ characters; each character of $$$s$$$ is either '(' or ')'. The third line contains the string $$$a$$$, consisting of exactly $$$n - 3$$$ characters; each character of $$$a$$$ is either '1' or '0'.

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 one integer: the minimum number of characters that need to be changed in $$$s$$$, or $$$-1$$$ if it is impossible.

✓	#	Name		✎	Rating↕	Solved↕
	1571E	Fix the String *special bitmasks dp greedy			1700	376
	1533H	Submatrices *special bitmasks data structures dp			unrated	92

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