
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 branded clothing with an annual fixed cost of $100,000. For every additional 1,000 items produced, the company invests $27,000. Let $x$ represent the annual production volume of this branded clothing, which is sold in its entirety. The revenue per 1,000 items, denoted as $R(x)$, is given by $R(x)=\begin{cases} & 10.8-\frac{{{x}^{2}}}{30},0 < x\leqslant 10 \\\\ & \frac{108}{x}-\frac{1000}{3{{x}^{2}}},x > 10 \\end{cases}$.

(1) Write the analytical expression for the annual profit $W$ (in ten thousand dollars) in terms of the annual production volume $x$ (in thousands of items).

(2) What annual production volume maximizes the profit for the company in the production of this branded clothing? (Note: Annual profit = Annual revenue - Annual cost)

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