
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 certain factory's annual fixed cost for a type of product is 2.5 million yuan. For every thousand units produced, denoted as $x$, an additional cost $C(x)$ (in million yuan) is incurred. When the annual output is less than 80 thousand units, $C(x) = \frac{1}{3}x^2 + 10x$ (in million yuan). When the annual output is not less than 80 thousand units, $C(x) = 51x + \frac{10000}{x} - 1450$ (in million yuan). The selling price per unit is 0.05 million yuan. Through market analysis, it is determined that the factory can sell all of its produced goods.

(1) Write the function expression for the annual profit $L(x)$ (in million yuan) in terms of the annual output $x$ (in thousand units);

(2) At what annual output (in thousand units) does the factory achieve the maximum profit for this product?

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