
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 $x$ units 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 units $x$ satisfy: $P^{2} = \frac{k}{x}$. The unit price for producing 100 units of this product is 50 ten thousand yuan. How many units should be produced to maximize the total profit?

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