
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. An additional investment of $100 is required for each instrument produced. The total revenue is given by the function: $R(x) = \begin{cases} 400x - \frac{1}{2}x^{2}, & 0 \leq x \leq 400, \\ 80,000, & x > 400, \end{cases}$ where $x$ is the monthly production volume. Profit = Total revenue - Total cost.

(1) Express the profit, $y$, as a function of the monthly production volume, $x$.
(2) For what value of $x$ does the company achieve maximum profit? What is the maximum profit?

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