OIM 2018 Problem 3

In a plane we have $n$ lines, no two of which are parallel or perpendicular, and no three of which are concurrent. A cartesian system of coordinates is chosen for the plane with one of the lines as the $x$ -axis. A point $P$ is located at the origin of the coordinate system and starts moving along the positive $x$ -axis with constant velocity. Whenever $P$ reaches the intersection of two lines, it continues along the line it just reached in the direction that increases its $x$ -coordinate. Show that it is possible to choose the system of coordinates in such a way that $P$ visits points from all $n$ lines.