
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 produces a type of electronic instrument with a fixed cost of $20,000. For each instrument produced, an additional investment of $100 is required. The total revenue is represented by the function: $_R_(x)= \begin{cases} 400x - \frac{1}{2}x^2, \quad 0 \leq x \leq 400, \\ 80,000, \quad x > 400, \end{cases}$ where $x$ denotes the monthly production volume.
(1) Express the profit $f(x)$ as a function of the monthly production volume $x$;
(2) For what value of $x$ does the company achieve the maximum profit? What is the maximum profit in dollars? (Total revenue = total cost + profit)

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