
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
Let  $T$  be a regular tetrahedron.  Let  $t$  be the regular tetrahedron whose vertices are the centers of the faces of  $T$ .  Let  $O$  be the circumcenter of either tetrahedron.  Given a point  $P$  different from  $O$ , let  $m(P)$  be the midpoint of the points of intersection of the ray  $\overrightarrow{OP}$  with  $t$  and  $T$ .  Let  $S$  be the set of eight points  $m(P)$  where  $P$  is a vertex of either  $t$  or  $T$ .  What is the volume of the convex hull of  $S$  divided by the volume of  $t$ ?

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