
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, based on past production and sales experience, has obtained the following relationship regarding production and sales: the fixed cost of the factory is ¥28,000, the production cost for each 100 units produced is ¥10,000, with a total cost function G(x) in ¥10,000 (total cost = fixed cost + production cost) where x represents the number of units produced in hundreds. If the sales revenue R(x) is given by 

$$
R(x) = 
\begin{cases} 
-0.4x^{2} + 4.2x & \text{for } 0 \leq x \leq 5, \\
11 & \text{for } x > 5
\end{cases}
$$

and the products produced achieve sales balance (meaning all produced items are sold), please solve the following problems:
(1) Determine the analytical expression for the profit function y = f(x) (profit = sales revenue - total cost).
(2) How many units of the new product should the factory produce to maximize profit?

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