
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
A sphere of radius \( R \) touches a plane \( \alpha \). Consider all possible spheres of radius \( r \) that touch this given sphere and the plane \( \alpha \). Find the geometric loci of the centers of these spheres and the points where they touch the plane \( \alpha \) and the given sphere.

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