
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-shaped object) has an inner radius of $3$ units and an outer radius of $6$ units and rests on a flat table. Determine the radius of the largest spherical ball that can rest on top of this torus while touching both the table surface and the torus itself. The form of the torus is by revolving a circle in the $xz$-plane that is centered at $(4.5, 0, 1)$ with a radius of $1.5$ around the $z$-axis. The sphere has its center on the $z$-axis and rests on either the table or the torus.

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