
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
Given a torus (doughnut-shaped object) with an inner radius of $3$ and an outer radius of $5$ sitting on a flat table. Determine the radius of the largest spherical ball that can be placed on top of the center of the torus such that the ball touches the horizontal plane. Assume the torus is formed by revolving the circle (center at $(4,0,1)$ and radius $1$) around the $z$-axis. The spherical ball has its center on the $z$-axis and touches either the table or the torus.

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