
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
A polyhedron with $n$ vertices, inscribed in a sphere of radius $R$, is called crystalline if a set of $n-1$ vertices of this polyhedron can be chosen such that all tetrahedra with vertices at any 4 points of this set are equi-volume. What is the maximum volume of a crystalline polyhedron? In the answer, indicate the found number, if necessary rounding it to two decimal places.

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