
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 produces a product x (in hundreds of units), with the total cost denoted by C(x) (in 10,000 yuan), where the fixed cost is 2 10,000 yuan. The cost increases by 1 10,000 yuan for every additional hundred units produced. The revenue R(x) is given by:
$$
R(x) = 
\begin{cases} 
4x - \frac {1}{2}x^{2} - \frac {1}{2}, & \text{if } 0 \leq x \leq 4, \\
7.5, & \text{if } x > 4 
\end{cases}
$$
(in 10,000 yuan), assuming the product achieves a break-even point.

(1) To ensure the factory does not incur a loss, within what range should the production quantity x be controlled?
(2) How many units should the factory produce annually to maximize profit?
(3) Determine the maximum profit per unit selling price of the product.

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