
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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A sphere with radius \( R \) has two cones inscribed in it with a common base; the vertices of the cones coincide with the opposite ends of the diameter of the sphere. The spherical segment containing the smaller cone has an arc in its axial section equal to \( \alpha^{\circ} \). Find the distance between the centers of the spheres inscribed in these cones.

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