
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 factory, inspired by Chang'e's flight to the moon on the Eastern Wind, has launched a new product under the brand "Jade Rabbit". The fixed cost for producing "Jade Rabbit" is 20,000 yuan, and an additional investment of 100 yuan is required for each "Jade Rabbit" produced. According to preliminary calculations, the total revenue satisfies the function $R(x) = \begin{cases} 400x - \frac{1}{2}x^2, (0 \leq x \leq 400) \\ 80000, (x > 400) \end{cases}$, where $x$ is the monthly production of "Jade Rabbit".
(1) Express the profit $f(x)$ as a function of the monthly production $x$;
(2) What monthly production value gives the maximum profit? What is the maximum profit? (Note: Total Revenue = Total Cost + Profit)

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