
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 has an annual fixed cost of $250 million for a certain product. When producing $x$ thousand units, an additional cost of $C(x)$ is required. When the annual production is less than $80$ thousand units, $C(x)=\frac{1}{3}x^2+10x$ (million dollars). When the annual production is not less than $80$ thousand units, $C(x)=51x+\frac{10000}{x}-1450$ (million dollars). Each unit of the product is sold at a price of $0.05$ million. According to market analysis, all the products produced by the factory can be sold.
(I) Write out the analytical expression of the annual profit $L(x)$ (million dollars) in terms of the annual production $x$ (thousand units);
(II) How many thousand units should be produced annually to maximize the profit from the production of this product?

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