
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 produces $x$ units of a product with a total cost of $c(x)=1200+ \frac{2}{75}{x}^{3}$ (in ten thousand yuan). The product's unit price $P$ (in ten thousand yuan) and the number of units $x$ satisfy the relationship: ${P}^{2}= \frac{k}{x}$. When 100 units of the product are produced, the unit price is 50 ten thousand yuan.

(1) Find the analytical expression for the total profit $L(x)$ (in ten thousand yuan) when producing $x$ units;

(2) Determine the number of units $x$ that maximize the total profit $L(x)$ (in ten thousand yuan), and calculate the maximum value (rounded to the nearest ten thousand yuan).

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