
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 octahedron, that  $C$  is the cube whose vertices are the centers of the faces of  $O$ , and that the ratio of the volume of  $O$  to that of  $C$  is  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime integers, find  $m+n$ .

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