
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
After graduating from college, Xiao Wang decided to start a business using his professional knowledge. After market research, it was determined that the fixed annual cost to produce a certain small electronic product would be 30,000 yuan. To produce $x$ tens of thousands of units, an additional variable cost of \(W(x)\) tens of thousands of yuan would be needed. When the annual output is less than 80,000 units, \(W(x)= \frac{1}{3}x^{2}+x\) tens of thousands of yuan. When the annual output is at least 80,000 units, \(W(x)=6x+ \frac{100}{x}-38\) tens of thousands of yuan. The selling price of each product is 5 yuan. According to market analysis, the goods produced by Xiao Wang can be sold out within the year.

(1) Write the function expression of the annual profit \(L(x)\) (in tens of thousands of yuan) as an analytical function of the annual output $x$ (in tens of thousands of units); (Note: Annual profit = Annual sales revenue - Fixed costs - Variable costs)

(2) At what annual output (number of tens of thousands of units) does Xiao Wang achieve the maximum profit for this product, and what is the maximum profit?

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