
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 certain tennis center plans to build several connected tennis courts on a 10,000 square meter land. Each tennis court has a construction area of 1,000 square meters. When the center builds $x$ ($x\in \mathbb{N}$) courts, the average construction cost per square meter can be approximated by the function $f(x)=800(1+\frac{1}{5}\ln x)$. Additionally, the center needs to pay a one-time environmental protection fee of $1,280,000$ yuan for this project.<br/>$(1)$ Write down the expression of the comprehensive cost $g(x)$ per square meter for the project when the tennis center builds $x$ ($x\in \mathbb{N}$) courts, and indicate its domain (comprehensive cost is the sum of construction cost and environmental protection fee);<br/>$(2)$ In order to minimize the comprehensive cost per square meter for the project, how many courts should the tennis center build?

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