
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, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra  $P_{i}$  is defined recursively as follows:  $P_{0}$  is a regular tetrahedron whose volume is 1. To obtain  $P_{i+1}$ , replace the midpoint triangle of every face of  $P_{i}$  by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of  $P_{3}$  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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