
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 (donut shape) has an inner radius of $3$ and an outer radius of $5$. It rests on a flat table. Determine the radius of the largest spherical ball that can be placed centered directly above the center of the torus so that the ball touches the horizontal plane (the $xy$-plane is the table). Suppose the torus in the $xz$-plane is originally centered at $(4, 0, 1)$ with a radius $1$ before being revolved 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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