Lame Queen Game

Code and Chef play a game on the set of all points with positive integer coordinates of the usual coordinate plane. Initially the lame queen stands in the point with coordinates (x0, y0) or shortly: in the point (x0, y0). Players alternate moves. Code moves first. During each move the player can move the queen from the point (x, y) to the point (x - a, y), (x, y - a) or (x - a, y - a) for some pos