Problem $3$: $(8$ points total. Show all work for partial credit $-$ use the back of this page as needed.$)$

Tags labeled with distinct ID#s are stored in $31$ bins. The ID#s are sorted in ascending order.

$\backslash$table$[[p=$ Bin#$,1,2,3,4,5,6,7,$dots,$1301,31],[$Tag ID #$,101,105,106,112,117,122,124,$dots,$197,199]]$

A robot searches the $31$ bins for a target tag ID# using the following algorithm.

Step $0$: Set k to $16.$

Step $1$: Move right by $k$ bins.

Step $2$: Pick up current tag.

Step $3$: Compare current tag ID# to target ID#$.$

Step $4$: If target ID# is equal to current tag ID#:

Turn on light to indicate that target was found. Stop.

$//$ comment: Else continue on to Step $5$

Step $5$: Put back the current tag.

Step $6$: Replace k with k$/2.$

Step $7$: If target ID# is less than current tag ID#:

Move left by $k$ bins

Else if target ID# is more than current tag ID#:

Move right by $k$ bins

Repeat from Step $2.$

$($a$)$ How many Steps are needed if the target tag ID# is equal to the tag ID# in Bin $10?$

$($b$)$ What is the maximum number of Steps needed for the robot to find a target tag in the bins? Give an example of a position that uses that maximum number of steps.