Consider the following algorithm segment.

For each positive integer $n,$ let $b_n$ be the number of iterations of the following while loop.

$)>(0$

Find a recurrence relation for $b_n.$

To answer this question, observe that for each integer $k3,$ndiv$(3)=$

Thus, when the algorithm segment is run for a particular $k$ and the while loop has iterated one time, the input to the next iteration is

It follows that the number of iterations of the loop for $k$ is more than the number of iterations for $[].$

In other words, for every integer $k,$ we have which of the following?

$b_k=1+b_{\frac{k}{3}}$

$b_k=1+b_{[\frac{k}{3}]}$

$b_k=1+b_{[\frac{k}{3}]}$

$b_k=3+b_{\frac{k}{3}}$

$b_k=3+b_{[\frac{k}{3}]}$

$b_k=3+b_{[\frac{k}{3}]}$

In addition, the initial values for the sequence $b_1,b_2,b_3,$dots are $b_1=$ and $b_2=$