
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 a market survey, it was found that the fixed annual cost to produce a certain product is 30,000 yuan. For every 10,000 units produced (x ten-thousands), an additional variable cost of W(x) in ten-thousands of yuan is incurred. When the annual production is less than 80,000 units, W(x) = $$\frac{1}{3}x^2 + x$$ (in ten-thousands of yuan); when the annual production is at least 80,000 units, W(x) = 6x + $$\frac{100}{x} - 38$$ (in ten-thousands of yuan). Market analysis indicates that if the unit price of the product is set at 5 yuan, the produced goods can be sold out within the year.

(1) Write the analytical expression for the annual profit function L(x) (in ten-thousands of yuan) in relation to the annual production x (in ten-thousands of units).
(2) Determine the production quantity at which the profit is maximized, and compute the maximum value.

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