
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
What is the locus of points $P$ in space such that three mutually perpendicular tangent planes to a given sphere with radius $r$ can be placed at $P$?

The three points of tangency determine a circle $k$ on the sphere. Determine:

a) the surface area of the smaller spherical cap corresponding to $k$,
b) the surface area of the conical frustum determined by $k$ and $P$.

How do these two surface areas compare to each other?

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