
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 \( R \) is inscribed in a spherical sector (see Fig. 12.239). Find the radius of the circle where the surfaces of the sphere and the sector touch, given that the central angle in the axial section of the spherical sector is \( \alpha \).

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