
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
If a certain company's fixed cost for producing a certain electronic instrument is $20,000, and an additional $100 is required for each instrument produced, it is known that the total revenue $R (\text{in units: yuan})$ as a function of monthly production $x$ (in units: units) satisfies the function: $R=\left\{\begin{array}{l}400x-\frac{1}{2}{x}^{2},0≤x≤500\\ 75000,x>500\end{array}\right.$.<br/>$(1)$ Express the profit $f\left(x\right) (\text{in units: yuan})$ as a function of the monthly production $x$;<br/>$(2)$ At what value of monthly production does the company achieve maximum profit? What is the maximum profit in yuan? (Total revenue $=$ Total cost $+$ Profit)

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