
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
In the tetrahedron \( S A B C \), the edges \( S A, S B, \) and \( S C \) are mutually perpendicular and have lengths \( a, b, \) and \( c \) respectively. 
a) Find the side length of the cube with vertex \( S \) that has the maximum volume and is entirely contained within the tetrahedron.
b) Determine the dimensions of the rectangular parallelepiped with vertex \( S \) that has the maximum volume and is entirely contained within the tetrahedron.

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