
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
Determine the radius of a sphere  $S$  that passes through the centroids of each face of a given tetrahedron  $T$  inscribed in a unit sphere with center  $O$ . Also, determine the distance from  $O$  to the center of  $S$  as a function of the edges of  $T.$ 

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