
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 regular tetrahedron has four equilateral triangular faces, with a sphere inscribed within it and another sphere circumscribed around it. For this tetrahedron, there are also four smaller spheres, each tangent externally to one face of the tetrahedron at its centroid, and to the circumscribed sphere. If a point $P$ is selected at random inside the circumscribed sphere, calculate the probability that $P$ lies inside one of the smaller spheres, when the radius of the circumscribed sphere is twice that of the inscribed sphere and the radius of each smaller sphere is the same as the radius of the inscribed sphere.

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