
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 order to further increase market competitiveness, Huawei plans to use new technology to produce a new mobile phone in the year 2023. Through market analysis, it is determined that a fixed cost of $2.5$ million is required to produce this mobile phone for the whole year. For each production of $x$ (thousand units) of the phone, an additional cost of $R(x)$ million is required, where $R(x)=\left\{\begin{array}{l}{10{x^2}+100x,0<x<40}\\{701x+\frac{{10000}}{x}-9450,x\geq 40}\end{array}\right.$. Market research shows that the selling price of each phone is $0.7$ million, and all phones produced within the year can be sold.

$(1)$ Find the analytical expression of the profit $W(x)$ in the year 2023 as a function of the annual output $x$ (in thousand units) (Profit = Revenue - Cost);

$(2)$ For what annual output $x$ (in thousand units) in 2023 will the company make the maximum profit? What is the maximum profit?

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