CZ-SK-MO 2024 Problem 5

Let $(a_k)^{\infty}_{k=0}$ be a sequence of real numbers such that if $k$ is a non-negative integer, then $$a_{k+1} = 3a_k - \lfloor 2a_k \rfloor - \lfloor a_k \rfloor.$$ Definitely all positive integers $n$ such that if $a_0 = 1/n$ , then this sequence is constant after a certain term.