CF1578H Higher Order Functions

Helen studies functional programming and she is fascinated with a concept of higher order functions — functions that are taking other functions as parameters. She decides to generalize the concept of the function order and to test it on some examples. For her study, she defines a simple grammar of types. In her grammar, a type non-terminal $ T $ is defined as one of the following grammar productions, together with $ \textrm{order}(T) $ , defining an order of the corresponding type: - "()" is a unit type, $ \textrm{order}(\textrm{"}\texttt{()}\textrm{"}) = 0 $ . - "(" $ T $ ")" is a parenthesized type, $ \textrm{order}(\textrm{"}\texttt{(}\textrm{"}\,T\,\textrm{"}\texttt{)}\textrm{"}) = \textrm{order}(T) $ . - $ T_1 $ "->" $ T_2 $ is a functional type, $ \textrm{order}(T_1\,\textrm{"}\texttt{->}\textrm{"}\,T_2) = max(\textrm{order}(T_1) + 1, \textrm{order}(T_2)) $ . The function constructor $ T_1 $ "->" $ T_2 $ is right-to-left associative, so the type "()->()->()" is the same as the type "()->(()->())" of a function returning a function, and it has an order of $ 1 $ . While "(()->())->()" is a function that has an order-1 type "(()->())" as a parameter, and it has an order of $ 2 $ . Helen asks for your help in writing a program that computes an order of the given type.

The single line of the input contains a string consisting of characters '(', ')', '-', and '>' that describes a type that is valid according to the grammar from the problem statement. The length of the line is at most $ 10^4 $ characters.

Print a single integer — the order of the given type.