Tyson is up to his beyblading ways again. In a beyblade battle, two players launch their beyblades (small spinning tops) into an arena to battle each other until one stops spinning.

This time, Tyson is up against a particularly defensive opponent, and needs to determine how close his beyblade can get to his opponent's since his own beyblade is equipped with a ranged attack. However, his opponent's beyblade has two defensive walls set up that try to keep other beyblades away.

Tyson has done some preliminary calculations and determined that the arena can be modeled as the $$$2D$$$ plane with the opponent's beyblade at the origin. The walls protecting the opponent's beyblade are both infinite lines that pass through the origin (forming an X-shape with the origin at the crossing). The first wall makes an angle of $$$\alpha$$$ degrees measured counterclockwise from the positive $$$x$$$-axis, and the second wall makes an angle of $$$\beta$$$ degrees (also measured counterclockwise from the positive $$$x$$$-axis). Tyson's beyblade begins at location $$$(p_x, p_y)$$$, and Tyson launches it at an angle of $$$\theta$$$ degrees (measured counterclockwise with respect to the vector extending from $$$(p_x,p_y)$$$ in the positive $$$x$$$ direction). When the beyblade collides with either wall, it reflects off the wall perfectly with angle of incidence equal to the angle of reflection, much like light bounces off of a mirror. Assuming that Tyson's beyblade keeps traveling for infinite time, what is the closest distance it will get to his opponent's beyblade, and how many total times will it bounce off of a wall?

It is guaranteed that $$$0 \leq \alpha \lt \beta \lt 180$$$, $$$\beta - \alpha \gt 10^{-2}, 180 - (\beta - \alpha) \gt 10^{-2}$$$, and that Tyson's beyblade starts at least distance $$$10^{-2}$$$ away from either wall. Furthermore, the minimum distance between Tyson's beyblade and the origin along its path of motion is at least $$$10^{-6}$$$ units.