
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]


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A certain factory's fixed annual cost for a type of product is 250 thousand yuan. To produce \(x\) thousand pieces, an additional cost of \(C(x)\) (unit: thousand yuan) is incurred. When the annual output is less than 80 thousand pieces, \(C(x) = \frac{1}{3}x^{2} + 10x\) (unit: thousand yuan). When the annual output is at least 80 thousand pieces, \(C(x) = 51x + \frac{10000}{x} - 1450\) (unit: thousand yuan). The selling price per product is 0.05 thousand yuan. Through market analysis, it is known that the factory can sell all the products it produces.

(1) Write down the function expression for annual profit \(L(x)\) (unit: thousand yuan) in terms of the annual output \(x\) (unit: thousand pieces);

(2) At what annual output (in thousand pieces) does the factory maximize its profit from producing this product?

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