
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 $Q_i$ defined recursively: $Q_0$ is a regular tetrahedron with a volume of 1. To obtain $Q_{i+1}$, replace the midpoint triangle of four chosen faces of $Q_i$ with an outward-pointing regular tetrahedron that has each midpoint triangle as a face and each new tetrahedron has side lengths equal to $\frac{2}{3}$ of the sides of the triangle it is built on. What is the volume of $Q_3$?

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