
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 $Q_{i}$ is defined recursively as follows: $Q_{0}$ is a regular tetrahedron whose volume is 1. To obtain $Q_{i + 1}$, replace the midpoint triangle of every face of $Q_{i}$ by an outward-pointing regular tetrahedron that has the midpoint triangle as a face, where the side length of each new tetrahedron is $\frac{1}{3}$ of its corresponding side on $Q_{i}$. Calculate the volume of $Q_4$.

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