
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
In a tetrahedron, given the same setup as in Example 7, let $\omega=\frac{1}{2}(\alpha+\beta+\gamma)$. The volume $V$ of the tetrahedron is then given by $V=\frac{a b c}{3} \sqrt{\sin \omega \cdot \sin (\omega-\alpha) \cdot \sin (\omega-\beta) \cdot \sin (\omega-\gamma)}$.

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