
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 company has a fixed cost of 20,000 yuan for producing a type of electronic instrument, and the cost increases by 100 yuan for each instrument produced. The total revenue is given by the function: $R(x)= \begin{cases} 400x- \frac {1}{2}x^{2}, & 0\leqslant x\leqslant 400\\ 80000, & x > 400\end{cases}$, where $x$ is the monthly production of the instrument. (Note: Total Revenue = Total Cost + Profit)

$(1)$ Express the profit $f(x)$ as a function of the monthly production $x$;  
$(2)$ At what monthly production value does the company achieve maximum profit? What is the maximum profit in yuan?

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