
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
In 2023, a certain travel company plans to develop a new amusement project in Fuqing Children's Park. It needs to invest a fixed cost of 300 million yuan for the whole year. If the project has x thousand visitors in 2024, it will need to add management and maintenance costs g(x) million yuan, where $g(x)=\left\{\begin{array}{l}20,0＜x≤4\\{x}^{2}+40x-100,x＞4\end{array}\right.$. The ticket price for this amusement project is 80 yuan per ticket.  
$(1)$ Find the function relationship $W(x)$ of the profit of this project in 2024 in terms of the number of visitors x (in thousand people) (Profit = Revenue - Cost).  
$(2)$ For how many visitors in 2024 will the project make the maximum profit? What is the maximum profit?

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