
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 for the Brazilian Olympic Games with an annual fixed cost of 2.5 million yuan. For every $x$ thousand units produced, an additional cost of $C(x)$ (in ten thousand yuan) is incurred. When the annual production is less than 80 thousand units, $C(x)=\frac{1}{3}x^2+10x$; when the annual production is not less than 80 thousand units, $C(x)=51x+\frac{10000}{x}-1450$. The selling price per product is 0.05 ten thousand yuan. Through market analysis, it is determined that all the products produced by the factory can be sold.

(1) Write the analytical expression of the annual profit $L$ (in ten thousand yuan) as a function of the annual production $x$ (in thousand units);

(2) At what annual production volume (in thousand units) does the factory maximize its profit from this product?

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