
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
(This question is worth 12 points)  
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 output is less than 80 thousand units, $C(x) = x^2 + 10x$ (in ten thousand yuan); when the annual output is not less than 80 thousand units, $C(x) = 51x - 1450$ (in ten thousand yuan). Through market analysis, it is determined that if each unit is sold for 500 yuan, the factory can sell all of its production for that year.  
(1) Write the analytical expression for the annual profit $L(x)$ (in ten thousand yuan) as a function of the annual output $x$ (in thousand units);  
(2) At what annual output (in thousand units) does the factory achieve the maximum profit from this product, and what is the maximum profit?

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