
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
Express the radius of the sphere that touches the planes of the side faces of a regular tetrahedron and is internally tangent to the circumsphere of the tetrahedron in terms of the dihedral angle between adjacent side faces and the base edge of the tetrahedron.

Apply the result to the case where the side faces of the tetrahedron are equilateral triangles.

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