
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 a sequence of polyhedra $P_{i}$ defined recursively, where $P_{0}$ is a regular tetrahedron with volume 2. For each face of $P_{i}$, replace the midpoint triangle by an outward-pointing regular tetrahedron with volume half of the midpoint triangle's face tetrahedron. Calculate the volume of $P_{2}$.

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