
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 $T$ is a regular tetrahedron, and $C$ is the cube whose vertices are the centers of the faces of $T$. If the side length of the tetrahedron is $s$, and it is given that $T$ and $C$ have a ratio of their volumes as $\frac{m}{n}$, where $m$ and $n$ are coprime, calculate $m+n$.

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