
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's annual fixed cost for producing a certain product is 2.5 million yuan. For every thousand units produced, an additional cost $C(x)$ is incurred. When the annual production is less than 80 thousand units, $C(x) = \frac{1}{3}x^2+10x$ (in ten thousand yuan); when the annual production is not less than 80 thousand units, $C(x) = 51x+ \frac{10000}{x} - 1450$ (in ten thousand yuan). The price of each product is 0.05 ten thousand yuan. Market analysis indicates that all the products manufactured by the factory can be sold out.

(1) Write the analytical expression for the annual profit $L(x)$ (in ten thousand yuan) as a function of the annual production $x$ (in thousand units).  
(2) How many thousand units should be produced annually to maximize the profit, and what is the maximum profit?

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