
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 regular tetrahedron $P_0$ with volume 1, a sequence of polyhedra $P_i$ is constructed as follows: For $P_{i+1}$, replace each face of $P_i$ with an outward-pointing regular tetrahedron whose base is the midpoint triangle of the face. If the volume of $P_4$ is $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$.

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