
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 $Q_i$, $Q_0$ is a regular tetrahedron with volume 2. To construct $Q_{i+1}$ from $Q_i$, replace the midpoint triangle of every face of $Q_i$ with an outward-pointing regular tetrahedron that has the midpoint triangle as a face. These new tetrahedra have side lengths $\frac{2}{3}$ of the sides of the tetrahedra to which they are attached. Starting from $Q_2$, each newly added tetrahedron has a volume twice that of the tetrahedron added in the corresponding position of the previous step. Find the volume of $Q_3$ and express it as a fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Then find $m+n$.

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