
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 cylinder is enclosed within a sphere. The sphere's radius is $S$ and the cylinder has a fixed volume $V$. The total surface area of the cylinder (including the two circular ends) is to be minimized. Given $S$, $V$, and that the volume of the cylinder is $\pi r^2 h = V$, find the ratio $\frac{H}{R}$ that minimizes the surface area.

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