
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 the same radius are stacked in the form of a triangular pyramid so that each sphere touches at least three others. Find the radius of the circumscribed sphere for the pyramid of spheres, given that the radius of the sphere inscribed in the center of the pyramid of spheres, touching six identical spheres, is $\sqrt{6}-1$.

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