
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 its past production and sales experience, has established the following statistical rules for production and sales: The total cost for producing x hundred units of a product is G(x) (in tens of thousands of yuan), with a fixed cost of 1.5 million yuan, and a production cost of 0.5 million yuan per hundred units (total cost = fixed cost + production cost). The sales income R(x) is defined as:
$$
R(x) = \begin{cases} 
-2x^2 + 21x + 1 & \text{for } 0 \leq x \leq 5 \\
56 & \text{for } x > 5 
\end{cases}
$$
Assuming that all products manufactured are sold (i.e., production is equal to sales), use the above statistical rules to complete the following questions:
(1) Write the analytical expression for the profit function y = f(x) (profit = sales income - total cost).
(2) Find the maximum profit the factory can obtain.

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