International Mathematical Olympiad 2017 Problem 1

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ by:

$$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise,} \end{cases} \quad \text{for each } n \geqslant 0.$$

Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.