
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
Construct a pyramid on each face of a regular octahedron with edge length one, where the pyramid is defined by the given face and the planes of the faces adjacent to it. What is the surface area and volume of the resulting star-shaped solid?

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