
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
A regular tetrahedron with a base edge length \( a \) and height \( m \) is inscribed with a cube in such a way that one face of the cube lies in the base plane of the tetrahedron, and the other four vertices of the cube are either a) on the side edges, or b) on the heights of the side faces. Calculate the volumes of these cubes as functions of \( a \) and \( m \). How do these volumes compare when \( m = a \)?

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