
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) having inner radius $3$ and outer radius $5$ sits on a flat table. What is 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? (If the $xy$-plane is the table, the torus is formed by revolving the circle in the $xz$-plane centered at $(4,0,1)$ with radius $1$ about the $z$-axis. The spherical ball has its center on the $z$-axis and rests on either the table or the donut.)

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