
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
Two cones have a common height; their vertices are at opposite ends of this height. The slant height of one cone is \( l \) and forms an angle \( \alpha \) with the height. The slant height of the other cone forms an angle \( \beta \) with the height. Find the volume of the common part of both cones.

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