RU-SPB 2020 Problem 96

The points $(1,1),(2,3),(4,5)$ and $(999,111)$ are marked in the coordinate system. We continue to mark points in the following way :

• If points $(a,b)$ are marked then $(b,a)$ and $(a-b,a+b)$ can be marked
• If points $(a,b)$ and $(c,d)$ are marked then so can be $(ad+bc, 4ac-4bd)$ .

Can we, after some finite number of these steps, mark a point belonging to the line $y=2x$ .