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.