
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]


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Three identical cylindrical surfaces with radius \(R\) and mutually perpendicular axes touch each other in pairs.

a) What is the radius of the smallest sphere that touches all three cylinders?

b) What is the radius of the largest cylinder that touches all three given cylinders, with its axis passing inside the triangle formed by the points of contact of the three given cylinders?

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