
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 volume of a regular pyramid is $V$. A plane is passed through the center of a sphere inscribed in the pyramid, parallel to its base. Find the volume of the pyramid that is cut off from the given pyramid by this plane, if the dihedral angle at the base is $\alpha$.

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