
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 octahedron, the centers of the eight faces are the vertices of a smaller octahedron. Find the ratio of the volume of the smaller octahedron to that of the larger one, and express the sum of the numerator and denominator of this ratio in simplest form.

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