
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 formed by revolving a circle in the $xz$-plane, which is centered at $(4,0,1)$ and has a radius of $2$, about the $z$-axis. The torus sits on a flat table represented by the $xy$-plane. Determine the radius of the largest spherical ball that can sit on top of the center of this torus such that the ball still touches the horizontal plane. The center of the spherical ball is on the $z$-axis.

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