
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
Given a cone with a base radius of $R$ and a height of $H$.
$(1)$ If there is an inscribed cylinder in the cone with a height of $x$, what value of $x$ makes the lateral surface area of the cylinder the largest? What is the largest lateral surface area?
$(2)$ If a plane divides the cone into a smaller cone and a frustum, when the volumes of the two geometric bodies are equal, find the ratio of the height of the smaller cone to the height of the frustum.

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