
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 certain product, requiring a fixed investment of \$(100\) million annually. Additionally, an investment of \$(1\) million is needed for each unit produced, with an annual production volume of \(x(x∈N^{∗})\) units. When \(x≤20\), the annual sales revenue is \((33x-x^{2})\) million dollars; when \(x>20\), the annual sales revenue is \$(260\) million. Let \(y\) represent the annual profit (in millions of dollars) earned by the factory from producing and selling this product. The functional relationship between \(y(\text{million dollars})\) and \(x(\text{units})\) is \_\_\_\_\_\_\_\_, and the factory's annual production volume is \_\_\_\_\_\_\_\_ units when the maximum annual profit is achieved (Annual Profit = Annual Sales Revenue - Annual Total Investment).

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