
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 bases of two cones, sharing a common vertex, lie in the same plane. The difference in their volumes is $V$. Find the volume of the smaller cone if the tangents drawn to the circle of its base from any point on the circle of the base of the larger cone form an angle $\alpha$.

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