
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
From which point on the hypotenuse of a right-angled triangle should perpendiculars be drawn to the legs so that the area of the resulting right-angled quadrilateral is maximized? If the shape is rotated around the leg, the right-angled triangle forms a cone, and the inscribed right-angled quadrilateral forms a cylinder; the question is: when will the surface area of the cylinder be the greatest?

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