
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 certain factory has a fixed annual cost of 2.5 million yuan for a type of product. For each \(x\) thousands of units produced, an additional cost \(C(x)\) is incurred. When the annual production is less than 80 thousand units, \(C(x) = \\dfrac{1}{3}x^2 + 10x\) million yuan. When the annual production is at least 80 thousand units, \(C(x) = 51x + \\dfrac{10000}{x} - 1450\) million yuan. Each unit of the product is sold at 0.05 million yuan. Market analysis assures that all the produced goods can be sold.
(1) Write an analytical expression for the annual profit \(L(x)\) (in million yuan) as a function of the annual production \(x\) (in thousand units).
(2) At what production level (in thousand units) does the factory maximize its profit from producing this product?

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