
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 $x$ pieces of a product is $c(x) = 1200 + \frac{2}{75}x^{3}$ (in ten thousand yuan). It is known that the unit price $P$ (in ten thousand yuan) of the product and the number of pieces $x$ satisfy: $p^{2} = \frac{k}{x}$. The unit price for producing 100 pieces of such a product is 50 ten thousand yuan.  
(1) Let the total profit when the output is $x$ pieces be $L(x)$ (in ten thousand yuan), find the expression for $L(x)$;  
(2) What is the output $x$ for which the total profit $L(x)$ (in ten thousand yuan) is maximized? And what is the maximum value (accurate to ten thousand yuan)?

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