
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 dodecahedron, start constructing outward-pointing regular tetrahedra on opposite pentagonal faces such that each face shares an edge with a face of $P_{i}$. Let $P_{0}$ be a regular dodecahedron with volume 1. To obtain $P_{i + 1}$, for each pair of opposite faces of $P_{i}$, construct two regular tetrahedra using the shared edge of the pentagon as their common base and pointing outward. The height of the tetrahedra is $r_{i}$, where $r_{i} = (\frac{1}{2})^i$. The volume of $P_{5}$ is $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Calculate $m+n$.

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