
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 torus sits on a flat table with an inner radius of $4$ and an outer radius of $6$. Determine the radius of the largest spherical ball that can be positioned at the center of the torus such that it still touches the horizontal table. Assume the torus is formed by revolving a circle in the $xz$-plane centered at $(5,0,1)$ with a radius $1$ about the $z$-axis. The spherical ball has its center on the $z$-axis.

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