
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 $x$ thousand units produced, an additional cost $C(x)$ 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 thousand units of the product is 50 million yuan. Market analysis shows that the factory can sell all the products it produces.

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

(Ⅱ) What is the annual output in thousand units for which the factory achieves the maximum profit from this product, and what is the maximum value?

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