
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
To save energy, protect the environment, and conserve materials, the "shape factor" $S$ of a building is defined as $S=\frac{F_{0}}{V_{0}}$, where $F_{0}$ is the area of the building exposed to the air (unit: square meters) and $V_{0}$ is the volume of the building (unit: cubic meters).  
$(1)$ If a cylindrical building has a base radius of $R$ and a height of $H$, with the exposed parts being the top and the lateral surface, find the "shape factor" $S$ of the building (express the result as an algebraic expression involving $R$ and $H).  
$(2)$ The "shape factor" of a building is defined as $f=\frac{L^{2}}{A}$, where $A$ is the base area of the building and $L$ is the perimeter of the base. Let $T$ be the total building area, which is the sum of the areas of each floor (the area of each floor is the base area). If $n$ is the number of floors in a dormitory building with a floor height of $3$ meters, then the "shape factor" of the dormitory building can be derived as $S=\sqrt{\frac{f \cdot n}{T}}+\frac{1}{3n}$. When $f=18$ and $T=10000$, find the value of the number of floors $n$ for which the "shape factor" $S$ of the dormitory building is minimized.

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