
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 that $O$ is a regular tetrahedron, and $C$ is the cube whose vertices are the centers of the faces of $O$, find the ratio of the volume of $O$ to that of $C$. Express your answer in the form of a fraction $\frac{p}{q}$, where $p$ and $q$ are coprime integers.

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