Consider a positive integer $a > 1$. If $a$ is not a perfect square then at the next move we add $3$ to it and if it is a perfect square we take the square root of it. Define the trajectory of a number $a$ as the set obtained by performing this operation on $a$. For example the cardinality of $3$ is $\{3, 6, 9\}$. Find all $n$ such that the cardinality of $n$ is finite.
The following part problems may attract partial credit.
$\textbf{(a)}$Show that the cardinality of the trajectory of a number cannot be $1$ or $2$.
$\textbf{(b)}$Show that $\{3, 6, 9\}$ is the only trajectory with cardinality $3$.
$\textbf{(c)}$ Show that there for all $k \geq 3$, there exists a number such that the cardinality of its trajectory is $k$.
$\textbf{(d)}$ Give an example of a number with cardinality of trajectory as infinity.
**Note:**In the following following problem I have tried to find the cases for the given options and prove the general case with induction but no possible idea came to my mind . It will be very nice if a rigorous answer is provided to all the given questions of the problem. Thank you.