
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 regular tetrahedron positioned in 3-space, the vertices are $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$. The centers of the four faces form a smaller tetrahedron. Calculate the ratio of the volume of this smaller tetrahedron to that of the larger tetrahedron and find the sum $m+n$ where $\frac{m}{n}$ is the simplified form of this ratio.

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