Fix the String

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.