KöMaL 2023 B5320

Given that $\frac{a_{n+3}}{a_{n+1}} + \frac{a_n}{a_{n+2}} = 2$ for each term of a sequence $a_n$, and the first three terms are $a_1=1$, $a_2=4$ and $a_3=2$, prove that $\frac{2^{2021}}{a_{2023}}$ is an integer.