
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 balls of the same radius are stacked in a triangular pyramid so that each ball touches at least three others. Find the radius of the sphere that circumscribes the pyramid of balls, given that the radius of the ball inscribed at the center of the pyramid of balls, touching six identical balls, is $\sqrt{2}-1$.

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