
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 chemical plant produces a certain product, with a production cost of 3 yuan per item. According to market research, when the factory price per item is $x$ yuan ($7 \leq x \leq 10$), the annual output is $(11 - x)^2$ ten thousand items. If all the products produced by the company are sold, then the company is considered to be operating normally. However, to protect the environment, the cost of pollution control is directly proportional to the output, with a constant proportionality coefficient $a$ ($1 \leq a \leq 3$).  
(1) Find the function relationship between the company's annual profit $L(x)$ and the factory price $x$.  
(2) When the factory price per item is set to how much yuan, the company's annual profit is maximized, and find the maximum profit.

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