
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
Consider a set of points that are inside or within one unit of a rectangular parallelepiped measuring $2$ by $3$ by $4$ units. Given that the volume of this set includes the original box, extensions of $1$ unit thickness from each face (without additional structures like spherical caps or cylinders), find the volume in terms of $\frac{m + n\pi}{p}$, if applicable.

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