
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 $4$ and outer radius $6$ sits on a flat table. What is the height of the tallest cylindrical object that can be placed inside the torus so that the cylinder's top surface is flush with the top surface of the torus? (If the $xy$-plane is the table, the torus is formed by revolving the circle in the $xz$-plane centered at $(5,0,1)$ with radius $1$ about the $z$-axis. The cylinder has its axis along the $z$-axis.)

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