
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 either inside or within one unit of a rectangular parallelepiped that measures 2 by 3 by 7 units. Calculate the volume of this set, presented as a simplified fraction of the form $\displaystyle \frac{m+n\pi}{p}$, where $m$, $n$, and $p$ are integers, and $n$ and $p$ are relatively prime. Find the sum $m+n+p$.

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