
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 total cost of producing $x$ items is given by $c(x) = 1200 + \frac{2}{75}x^{2}$ (in ten thousand yuan). It is known that the unit price $P$ (in ten thousand yuan) of the product and the number of items $x$ satisfy: $P= \frac{k}{x}$. The unit price for producing 100 items of such a product is 50 ten thousand yuan.  
(1) Let the total profit when producing $x$ items be $L(x)$ (in ten thousand yuan), find the expression for $L(x)$.  
(2) At what production quantity $x$ is the total profit $L(x)$ (in ten thousand yuan) maximized? And find the maximum value (accurate to ten thousand yuan).

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