
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 25 million yuan. For the production of x thousand units, additional costs $C(x)$ are incurred. When the annual production is less than 80 thousand units, $C(x)=\frac{1}{3}x^{2}+10x$ (ten thousand yuan); when the annual production is not less than 80 thousand units, $C(x)=51x+\frac{10000}{x}-1450$ (ten thousand yuan). Each unit is priced at $0.05$ million yuan, and through market analysis, it is determined that all products produced by the factory can be sold.

1. Write out the analytical expression of the annual profit $L(x)$ (ten thousand yuan) as a function of the annual production $x$ (thousand units).
2. What is the annual production (in thousand units) when the profit from producing this product is maximized?

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