
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 for producing a type of electronic instrument, and the variable cost per unit is ¥100. 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 volume.

(1) Express the profit as a function of the monthly production volume $f(x)$;

(2) At what monthly production volume is the company's profit maximized? What is the maximum profit? (Total Revenue = Total Cost + Profit)

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