CF1095E Almost Regular Bracket Sequence

You are given a bracket sequence $ s $ consisting of $ n $ opening '(' and closing ')' brackets. 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 "()()", "(())" are regular (the resulting expressions are: "(1)+(1)", "((1+1)+1)"), and ")(" and "(" are not. You can change the type of some bracket $ s_i $ . It means that if $ s_i = $ ')' then you can change it to '(' and vice versa. Your task is to calculate the number of positions $ i $ such that if you change the type of the $ i $ -th bracket, then the resulting bracket sequence becomes regular.

The first line of the input contains one integer $ n $ ( $ 1 \le n \le 10^6 $ ) — the length of the bracket sequence. The second line of the input contains the string $ s $ consisting of $ n $ opening '(' and closing ')' brackets.

Print one integer — the number of positions $ i $ such that if you change the type of the $ i $ -th bracket, then the resulting bracket sequence becomes regular.