Counting Graph Intersections

The natural constant $e = 2.7182818284590452353602874\cdots$ is the base of the natural logarithm. It is known that it can be written as follows.

$e = \lim_{t \rightarrow \infty} \left(1+\frac{1}{t}\right)^t = \sum_{k=0}^{\infty} \frac{1}{k!}$

The graph of a function $f : X \rightarrow Y$ with domain $X$ and codomain $Y$ is the set of ordered pairs of inputs and outputs, $\{(x, f(x)) : x \in X\}$. If $X$ and $Y$ are both sets of real numbers, the points of the graph can be drawn on a two-dimensional coordinate plane.

The functions in this problem have the form y=e?(ax). Here ? is one of the multiplication operator *, the division operator /, or the exponentiation operator ^, and a is an integer. If the operator is /, then a is guaranteed not to be 0.

The domain is all real numbers ($\mathbb{R}$) when ? is * or ^, and all real numbers except 0 ($\mathbb{R} \setminus \{0\}$) when ? is /. The codomain is the set of real numbers ($\mathbb{R}$) in every case. So every function in this problem can be drawn on the coordinate plane.

The figure shows graphs of a few functions for different operators ? and integers a.

Write a program that counts the distinct intersection points when the graphs of all the given functions are drawn on the coordinate plane. A point is an intersection if it belongs to at least two of the graphs. If a point belongs to three or more graphs, it is counted once. The figure has 2 intersection points.

The first line contains the number of functions $n$ ($1 \le n \le 300{,}000$). Each of the next $n$ lines describes one function with an operator ? (one of *, /, ^) and an integer a ($-10^6 \le a \le 10^6$), separated by a single space. All given functions are distinct. If ? is /, then a is not 0.

Print the number of intersection points when the graphs of all the given functions are drawn on the coordinate plane.