A14480 | 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.