
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 cube, its center square of each face 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 cube whose volume is 1 cubic unit. To obtain $Q_{i+1}$, replace the center square of every face of $Q_i$ by an outward-pointing regular cube that has the center square as one face. The volume of $Q_3$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

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