Tendency for cycles to have the same length in Collatz type maps

Question

I was looking at modified versions of the Collatz map of the form $$ C_k(n) = \begin{cases} n/2 \text{ if } n \text{ is even} \\ 3n + k \text{ if } n \text{ is odd} \end{cases} $$ I noticed that for values of $k$ where there are many different cycles, almost all the cycles have length equal to multiples of a certain number. For example, if $k=499$, there are $53$ distinct cycles with smallest element less than $100000$, and of those, $41$ have a length of $26$, and $8$ more have lengths that are a multiple of $26$, and if $k = 1909$, then there are $87$ distinct cycles, and of those, $70$ have length $19$, and $17$ more have lengths that are a multiple of $19$. I wrote some code to look for values of k that had many different cycles, and it seemed like this was a common pattern. What causes these maps to have cycles with multiples of the same length?