
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 factory produces a certain product with an annual fixed cost of $250 million. When producing $x$ (where $x$ is a natural number) thousand units, an additional cost of $C(x)$ million is required. When the annual production is less than 80 thousand units, $C(x) = \frac{1}{3}x^2 + 10x$ (in million yuan); when the annual production is not less than 80 thousand units, $C(x) = 51x + \frac{10000}{x} - 1450$ (in million yuan). Through market analysis, it is found that when each unit is sold at 500 yuan, all products produced by the factory can be sold within the year.

(1) Write the analytical expression of the annual profit $L$ (in million yuan) in terms of the annual production $x$ (in thousand units);

(2) How many thousand units should be produced annually to maximize the profit from producing this product?

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