
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]


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

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