
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
Given a regular tetrahedron with edge length $a$, find the radius of a sphere centered at the center of the tetrahedron such that the total volume of the part of the tetrahedron outside the sphere and the part of the sphere outside the tetrahedron is minimized.

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