KURSCHAK 1989 Problem 3

We play the following game in a Cartesian coordinate system in the plane. Given the input $(x,y)$ , in one step, we may move to the point $(x,y\pm 2x)$ or to the point $(x\pm 2y,y)$ . There is also an additional rule: it is not allowed to make two steps that lead back to the same point (i.e, to step backwards).

Prove that starting from the point $\left(1;\sqrt 2\right)$ , we cannot return to it in finitely many steps.