Middle European Mathematical Olympiad 2018 Problem T-7

Let $a_1, a_2, a_3, \ldots$ be the sequence of positive integers such that $$a_1 = 1 \quad \text{and} \quad a_{k+1} = a_k^3 + 1, \text{ for all positive integers } k.$$

Prove that for every prime number $p$ of the form $3\ell + 2$, where $\ell$ is a non-negative integer, there exists a positive integer $n$ such that $a_n$ is divisible by $p$.