
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Circle $C$ has a radius of $36$. Circle $D$ has an integer radius $s$ such that $1 \leq s < 36$. Circle $D$ remains internally tangent to circle $C$ as it rolls once around the circumference of circle $C$. The two circles have the same points of tangency at the beginning and end of circle $D$'s trip. Circle $D$ rolls around circle $C$ such that the total trip can be divided evenly into a whole number of rotations around circle $C$. How many possible values can $s$ have?
A) 3
B) 5
C) 6
D) 8
E) 10

Present the answer in LaTex format: \boxed{Your answer}
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