
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 tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. There are eight smaller spheres, each tangent externally to one subdivided equilateral triangular region (with one vertex at the center of the tetrahedron face and the other vertices as the midpoints of the face's edges) of each face of the tetrahedron, and to the circumscribed sphere. A point $P$ is selected at random inside the circumscribed sphere. Compute the probability that $P$ lies inside one of the eight small spheres.

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