
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
The 19th Asian Games will be held in Hangzhou, China from September 23 to October 8, 2023. The mascots of the Asian Games, Chenchen, Congcong, and Lianlian, known as the "Jiangnan Yi Combination," are beloved by many. After conducting market research, a manufacturer found that the annual fixed cost for producing the "Jiangnan Yi Combination" small toys is 60,000 yuan. For each x thousand sets of the product produced, an additional variable cost W(x) in thousand yuan is required. When the annual production is less than 120,000 sets, W(x) = 0.5x^2 + 4x. When the annual production is at least 120,000 sets, W(x) = 11x + 100/x - 39. The selling price of each product is 10 yuan. Assume that the annual sales volume of the product is equal to the production volume of that year.

$(1)$ Write the analytical expression of the function of annual profit L(x) (in thousand yuan) with respect to the annual production volume x (in thousand sets). (Note: Annual profit = Annual sales revenue - Fixed cost - Variable cost)

$(2)$ For how many thousand sets of annual production volume will the annual profit be maximized? What is the maximum annual profit?

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