
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 branch company sells a certain brand of products, with a cost of 3 yuan per product. Additionally, each product incurs a management fee of \\(a\\) yuan \\((3 \leqslant a \leqslant 5)\\) to the head office. It is estimated that when the selling price of each product is \\(x\\) yuan \\((9 \leqslant x \leqslant 11)\\), the annual sales volume will be \\((12-x)^{2}\\) ten thousand units.

    \\((1)\\) Find the function relationship between the branch company's annual profit \\(L\\) (in ten thousand yuan) and the selling price per product \\(x\\).

    \\((2)\\) At what selling price per product will the branch company's annual profit \\(L\\) be maximized, and what is the maximum value of \\(L\\), denoted as \\(Q(a)\\)?

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