
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 sphere with a radius of $\frac{\sqrt{22}}{2}$ and center $O$. The center $O$ is also the centroid of a regular tetrahedron $\Gamma$, and the total length of the curves formed by the intersection of the sphere with the four faces of the tetrahedron $\Gamma$ is $8 \pi$. Find the volume of the tetrahedron $\Gamma$.

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