
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 enterprise in our country plans to use new technology to produce a new mobile phone in the year 2023 in order to further increase its market competitiveness. Through market analysis, it is determined that a fixed cost of $2.5$ million is required for the production of this mobile phone for the whole year. For each production of $x$ (thousand units) of the phone, an additional variable cost of $R(x)$ million is required, where $R(x)=\left\{\begin{array}{l}{10x^2+200x+1000, 0<x<40}\\{801x+\frac{10000}{x}-8450, x\geq40}\end{array}\right.$. According to market research, the selling price of each phone is $0.8$ million, and all phones produced within the year can be sold in the same year. (Profit = Revenue - Fixed Cost - Variable Cost).

$(1)$ Find the function relationship $W(x)$ (in million yuan) of profit in the year 2023 with respect to the annual output $x$ (thousand units).

$(2)$ When the output in 2023 is how many (thousand units), will the enterprise make the maximum profit? What is the maximum profit?

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