
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 an annual fixed cost of $100,000 for producing a certain brand of clothing, with an additional investment of $27,000 required for each 1,000 pieces produced. Let $x$ be the number of thousands of pieces produced and sold by the company in a year, with a sales revenue of $R(x)$ per thousand pieces, where $R(x)$ is defined as follows:
$$
R(x) =
\begin{cases}
10.8 - \frac{1}{30}x^2, & 0 < x \leq 10 \\
\frac{108}{x} - \frac{1000}{3x^2}, & x > 10
\end{cases}
$$
(1) Write the analytical expression for the annual profit $W$ (in ten thousand dollars) as a function of the annual production $x$ (in thousands of pieces).

(2) Determine the annual production (in thousands of pieces) that maximizes the company's annual profit in the production of this brand of clothing. (Note: Annual profit = Annual sales revenue - Annual total cost)

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