
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 height of a cone is \( H \), and the angle between the slant height and the altitude is \( \alpha \). Inside this cone, another cone is inscribed such that the vertex of the second cone coincides with the center of the base of the first cone, and the corresponding slant heights of both cones are mutually perpendicular. Find the volume of the inscribed cone.

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