
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
Consider a torus (donut shape) with an inner radius of $3$ and an outer radius of $5$ resting on a flat table. Determine the radius of the largest spherical ball that can be positioned atop the center of this torus such that the ball touches the horizontal plane below. Assume the torus is generated by revolving a circle in the $xz$-plane centered at $(4,0,1)$ with a radius $1$ about the $z$-axis, and the sphere is centered along the $z$-axis.

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