ISL 2017 N1

The sequence $a_{0}, a_{1}, a_{2}, \ldots$ of positive integers satisfies $$a_{n+1}=\left\{\begin{array}{ll} \sqrt{a_{n}}, & \text { if } \sqrt{a_{n}} \text { is an integer } \\ a_{n}+3, & \text { otherwise } \end{array} \text { for every } n \geqslant 0\right.$$ Determine all values of $a_{0}>1$ for which there is at least one number $a$ such that $a_{n}=a$ for infinitely many values of $n$. (South Africa)