
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 the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $2$ by $3$ by $4$ units. Along with the parallelepiped, half-spheres of radius $1$ unit are placed at each vertex and full cylinders of radius $1$ unit are placed along each edge of the parallelepiped. Calculate the total volume of this configuration if the volume is expressed as $\frac{q + r\pi}{s}$, where $q, r,$ and $s$ are positive integers, with $r$ and $s$ being relatively prime. Find $q + r + s$.

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