Question 25
The sequence $a_{0}, a_{1}, a_{2}, \ldots$ of positive integers satisfies
$$
a_{n+1}=\left\{\begin{array}{ll}
\sqrt{a_{n}}, & \text { if } \sqrt{a_{n}} \text { is an integer } \\
a_{n}+3, & \text { otherwise }
\end{array} \quad \text { for every } n \geqslant 0\right.
$$
Determine all values of $a_{0}\gt 1$ for which there is at least one number $a$ such that $a_{n}=a$ for infinitely many values of $n$.
$$
a_{n+1}=\left\{\begin{array}{ll}
\sqrt{a_{n}}, & \text { if } \sqrt{a_{n}} \text { is an integer } \\
a_{n}+3, & \text { otherwise }
\end{array} \quad \text { for every } n \geqslant 0\right.
$$
Determine all values of $a_{0}\gt 1$ for which there is at least one number $a$ such that $a_{n}=a$ for infinitely many values of $n$.