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{- \gt }\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.

海伦正在学习函数式编程，她对高阶函数（即以其他函数作为参数的函数）这一概念非常着迷。她决定将函数“阶数”的概念进行推广，并在一些示例上加以验证。

为开展这项研究，她定义了一种简单的类型文法。在此文法中，类型非终结符 $T$ 由以下文法规则之一定义，并同时定义其对应类型的阶数 $\textrm{order}(T)$：

• "()" 是单位类型，$\textrm{order}(\textrm{"}\texttt{()}\textrm{"}) = 0$。
• "( " $T$ ")" 是加括号的类型，$\textrm{order}(\textrm{"}\texttt{(}\textrm{"}\,T\,\textrm{"}\texttt{)}\textrm{"}) = \textrm{order}(T)$。
• $T_1$ "\->" $T_2$ 是函数类型，\textrm{order}(T_1\,\textrm{"}\texttt{- \gt }\textrm{"}\,T_2) = \max(\textrm{order}(T_1) + 1, \textrm{order}(T_2))。函数构造符 $T_1$ "\->" $T_2$ 是右结合的，因此类型 "()->()->()" 等价于 "()->(()->())"（即返回一个函数的函数），其阶数为 $1$；而 "(()->())->()" 则是一个以阶数为 $1$ 的类型 "(()->())" 为参数的函数，其阶数为 $2$。

海伦请你帮忙编写一个程序，用于计算给定类型的阶数。

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.

输入仅有一行，包含一个由字符 '('、')'、'-' 和 '>' 组成的字符串，该字符串根据题目描述中的语法规则是一个合法的类型。该行长度至多为 $10^4$ 个字符。

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

输出一个整数——给定类型的阶数。