BALTIC-WAY 2024 Problem 18

An infinite sequence $a_{1}, a_{2}, \ldots$ of positive integers is such that $a_{n} \geq 2$ and $a_{n+2}$ divides $a_{n+1}+a_{n}$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.