
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
When you cut along a generatrix of a cylinder, you can unfold the lateral surface onto a plane. The resulting rectangle is known as the unfolded lateral area of the cylinder, where the length and width of the rectangle correspond to the circumference of the cylinder's base and the height (generatrix length), respectively. Therefore, the lateral area $S$ of the cylinder is given by $S = 2\pi rl$, where $r$ is the radius of the cylinder's base and $l$ is the length of the generatrix. Now, consider a cone with base radius $R$ and height $H$, inside which there is an inscribed cylinder of height $x$.  
(1) Find the lateral area of the cylinder.  
(2) For what value of $x$ is the lateral area of the cylinder maximized?

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