For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as

$$a_{n+1} =

\begin{cases}

\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\

a_n + 3 & \text{otherwise.}

\end{cases}

$$

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