
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 incurs a fixed cost of 5000 yuan when producing a certain product. For every 100 units produced, the direct cost increases by 2500 yuan. The annual demand for this product in the market is 500 units. The revenue function is $R(x) = 5x - x^2$ (in ten thousand yuan) where $0 \leq x \leq 5$, and $x$ is the quantity of the product sold (in hundreds of units).

(1) Express the profit as a function of the annual production volume.
(2) At what annual production volume does the company achieve maximum profit?
(3) At what annual production volume does the company break even?

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