
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
Consider a regular tetrahedron $T$ with side length $s$. Let $C$ be a cube where each vertex is the centroid of one of the faces of $T$. Find the ratio of the volume of $T$ to that of $C$, given as a simplified fraction $\frac{m}{n}$ where $m$ and $n$ are coprime integers. Determine $m+n$.

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