KöMaL 2018 B4950

Let $F_n$ denote the $n$th Fibonacci number ($F_1=F_2=1$, $F_{n+2}= F_{n+1}+F_n$), and define the sequence $a_0,a_1,a_2,\dots$ with the following recurrence relation: let $a_0=2018$, and for all $k\ge 0$ let $a_{k+1}=a_k+F_n$, where $F_n$ is the largest Fibonacci number less than $a_k$. Will there be any Fibonacci number in the sequence $(a_k)$?