
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 $7$. It sits 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 still touches the horizontal plane. (In this scenario, the torus is formed by revolving a circle in the $xz$-plane centered at $(5,0,2)$ with a radius of $2$ about 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}
user