Školjka

Let $a_0,a_1,a_2,\ldots$ be a sequence of integers and $b_0,b_1,b_2,\ldots$ be a sequence of positive integers such that $a_0=0,a_1=1$, and \[ a_{n+1} = \begin{cases} a_nb_n+a_{n-1} & \text{if $b_{n-1}=1$} \\ a_nb_n-a_{n-1} & \text{if $b_{n-1}>1$} \end{cases}\qquad\text{for }n=1,2,\ldots. \]for $n=1,2,\ldots.$ Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to $2017$.