For each integer $a_0 >$ 1, define the sequence $a_0, a_1, a_2, \cdots$ by: $$ a_{n+1} = \left\{ \begin{array}{ll} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer}\\ a_n + 3 & \text{otherwise} \end{array} \right. $$ For all $n \ge 0$. Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.