RU-SPB 2020 Problem 116

The sequence $a_n$ is given as $$a_1=1, a_2=2 \;\; \text{and} \;\;\; a_{n+2}=a_n(a_{n+1}+1) \forall n\geq 1$$ Prove that $a_{a_n}$ is divisible by $(a_n)^n$ for $n\geq 100$ .