
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 has a fixed cost of 0.5 ten thousand yuan for producing a type of machine. However, for every 100 units produced, an additional variable cost of 0.25 ten thousand yuan is incurred. The annual demand for this product in the market is 500 units, and the revenue function (in ten thousand yuan) is $R(x) = 5x - \frac{1}{2}x^2 (0 \leqslant x \leqslant 5)$, where $x$ is the quantity of the product produced (in hundreds of units).
1. Express the profit as a function of the production quantity.
2. At what annual production quantity does the company achieve maximum profit?

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