
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 triangle in space, its midpoint triangle is formed by joining the midpoints of its sides. From an initial polyhedron $P_0$ which is a regular tetrahedron with volume 1, define a sequence of polyhedra $P_i$ as follows: $P_{0}$ is given, and $P_{i + 1}$ is obtained by replacing each edge of every face of $P_i$ with two outward-pointing regular tetrahedra, each having one of the edge's endpoints and the midpoint of the edge as vertices with volume scaling as $\left(\frac{1}{3}\right)^3$ of the tetrahedron being replaced. Compute the volume of the polyhedron $P_{3}$ and if it is $\frac{m}{n}$, find $m + n$ where $m$ and $n$ are relatively prime.

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