
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
The centers of two spheres with different radii lie in a plane \( \Pi \). Find the locus of points \( X \) in this plane such that a plane passing through \( X \) can be tangent to the spheres: 

a) internally (the spheres are on opposite sides of the plane).

b) externally (the spheres are on the same side of the plane).

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