Moving Point in a set of non-concurrent lines

Question

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