
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
Given a tetrahedron with face areas \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\), and the radius \(R\) of the inscribed sphere, obtain the formula for the volume of the tetrahedron.

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