
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. Each additional instrument produced requires an added investment of $100. The total revenue is represented by the following function: $R(x) = \begin{cases} 400x - \frac{1}{2}x^2 & \text{for } 0 \leq x \leq 400 \\ 80,000 & \text{for } x > 400 \end{cases}$, where $x$ represents the monthly production volume.
(I) Express profit as a function of the monthly production volume $x$ (denoted as $f(x)$);
(II) What value of the monthly production volume yields the maximum profit for the company? What is the maximum profit in dollars? (Note: Profit = Total Revenue - Total Cost)

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