
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 is created by revolving a circle in the $xz$-plane centered at $(4,0,1)$ with radius $1$. The torus has an inner radius of $3$ and outer radius of $5$ and sits on a flat table represented by the $xy$-plane. Determine the radius of the largest spherical ball that can be placed on top of the center of the torus so that the ball still touches the horizontal plane.

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