
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 $3$ by $3$ by $6$ units. Determine the volume of this set, which should include the main box, $6$ surrounding boxes (one for each face with 1 unit extension), $8$ quarter-spherical regions on each vertex (with radius $1$), and $12$ quarter-cylindrical regions along each edge (also of radius $1$). Express the volume in the simplest fractional form possible.

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