
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
Given that a company's annual fixed cost for producing a certain product is 100,000 yuan, an additional cost of 27,000 yuan is incurred for every thousand pieces produced. Suppose the company produces $x$ thousand pieces of this product within a year and sells them all, with a revenue of $f(x)$ ten-thousand yuan per thousand pieces, where
$$
f(x) = 
\begin{cases} 
10.8 - \frac{1}{30}x^2 & (0 < x \leq 10) \\ 
\frac{108}{x} - \frac{1000}{3x^2} & (x > 10)
\end{cases}.
$$
(I) Write the function expression for the annual profit $P$ (ten-thousand yuan) in relation to the product's annual output $x$ (thousand pieces);
(II) At what annual output $x$ (thousand pieces) does the company achieve maximum annual profit for this product? (Note: Annual profit = Annual sales revenue - Annual total cost)

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