Let $\operatorname{rad}(k)$ denote the product of prime divisors of a natural number $k$ (define $\operatorname{rad}(1)=1$). A sequence $(a_n)$ is defined by setting $a_1$ arbitrarily, and $a_{n+1}=a_n+\operatorname{rad}(a_n)$ for $n\ge1$. Prove that the sequence $(a_n)$ contains arithmetic progressions of arbitrary length.