
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
Ten spheres of identical radius are stacked in the shape of a triangular pyramid such that each sphere touches at least three others. Find the radius of a sphere inscribed in the center of the pyramid of spheres, which touches six identical spheres, if the radius of the sphere into which the pyramid is inscribed is equal to $\sqrt{6}+1$.

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