KöMaL 2019 B5059

For a positive integer $c$, define the sequence $\{ a_n\}$ by the following recurrence relation: $a_0=c$ and $a_{n+1} = \big[ a_{n} + \sqrt{a_{n}}\,\big]$ for $n \ge 0$. Prove that if $2019$ occurs as a term of the sequence, then no preceding term is a perfect square, but there are infinitely many perfect squares among the following terms.