KöMaL 2025 A914

We write the ordered pair $(1,0)$ on the board. In one step, if the ordered pair $(a,b)$ is on the board, we erase it and replace it with either $(a+b,b)$ or $(2a+b,a+b)$. (For example, after two steps, the pair on the board may be one of $(1,0)$, $(2,1)$, $(3,1)$, or $(5,3)$.) Prove that after $n$ steps, the ordered pair on the board can take exactly $2^n$ different values.