
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's fixed cost for producing a model of small household air purifiers is 20,000 yuan. Each additional purifier produced incurs an added cost of 200 yuan. Market research indicates that the total revenue (in yuan) R(x) from selling these air purifiers can be described by the function
$$ R(x) = \begin{cases} 500x - \frac{1}{2}x^2, & 0 \leq x \leq 400 \\ 84500 + 100x, & x > 400 \end{cases} $$
where x is the monthly production (in units) of the air purifiers.

(1) Express the company's monthly profit function $f(x)$ in terms of the monthly production $x$.
(2) For what value of monthly production $x$ is the company's monthly profit maximized? Also, calculate the maximum monthly profit.

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