MEMO 2018 Problem 7

Let $a_1,a_2,a_3,\cdots$ be the sequence of positive integers such that $$a_1=1 , a_{k+1}=a^3_k+1,$$ for all positive integers $k.$ Prove that for every prime number $p$ of the form $3l +2,$ where $l$ is a non-negative integer ,there exists a positive integer $n$ such that $a_n$ is divisible by $p.$