
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
There is a rectangular box with dimensions $m \times n \times p$ and another rectangular box with dimensions $(m+2) \times (n+2) \times (p+2)$, where $m$, $n$, and $p$ are positive integers and $m \leq n \leq p$. The volume of the former is half of the volume of the latter. Find the maximum value of $p$.

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