Bessie has just purchased a new laptop computer, but she unfortunately finds herself unable to type well, given the size of her large hooves relative to the small keyboard. Bessie has just attempted to type in one of her favorite patterns -- a balanced string of parentheses. However, she realizes that she might have mis-typed one character, accidentally replacing ( with ) or vice versa. Please help Bessie compute the number of locations in the string such that reversing the single parenthesis at that location would cause the entire string to become balanced.

A string of parentheses is "balanced" if there are the same total number of ('s and )'s, and for any prefix of the string there are at least as many ('s as )'s.

Line 1: A string of parentheses of length N (1 <= N <= 100,000).

The number of positions within the input string (if any) such that reversing the parenthesis at that single position would cause the entire string to become balanced.

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