
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 produces a certain product with an annual fixed cost of $100,000. For every $1,000$ units produced, an additional investment of $27,000$ is required. Let $x$ denote the number of thousands of units produced and sold by the company within a year, and let $R(x)$ be the revenue per thousand units sold, where $R(x) = \begin{cases} 10.8 - \frac{1}{30}x^2, & (0 < x \leq 10) \\ \frac{108}{x} - \frac{1000}{3x^2}, & (x > 10) \end{cases}$.

1. Write the analytical expression for the annual profit $W$ (in ten thousand dollars) in terms of the annual production $x$ (in thousands of units).
2. What is the annual production quantity (in thousands of units) that maximizes the company's annual profit in the production and sales process of this product?

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