CMO 2020 Problem 5

Given any positive integer $c$ , denote $p(c)$ as the largest prime factor of $c$ . A sequence $\{a_n\}$ of positive integers satisfies $a_1>1$ and $a_{n+1}=a_n+p(a_n)$ for all $n\ge 1$ . Prove that there must exist at least one perfect square in sequence $\{a_n\}$ .