
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 regular tetrahedron $P_0$ with a volume of $1$, we construct a sequence of polyhedra $Q_{i}$ where $Q_{i+1}$ is obtained by attaching outward-pointing regular tetrahedra to each face of $Q_i$. Each attached tetrahedron's face coincides with the midpoint triangle of a face of $Q_i$, and the size of each succeeding tetrahedron is $\frac{1}{3}$ the size of their predecessors on $Q_0$. Find the volume of $Q_4$.

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