A sequence defined by $a_1=2,a_2=\frac{4}{5}$ and $a_{n+2}=a_n^2-2a_{n+1}-4(n\ge 1)$

Question

Let $\{a_n\}$ be a sequence satisfying $a_1=2,a_2=\frac{4}{5}$ and $a_{n+2}=a_n^2-2a_{n+1}-4(n\ge 1)$. Prove that

1. $|a_n|\le 4$;
2. $\prod_{k=1}^n |a_{2k}|\le \frac{5}{\sqrt{11}}$.

By graphing recursively, one can suspect $$2|a_n-4|<a_{n+1}<\frac{1}{4}a^2_{n},$$ but which is still hard to prove.