
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 company produces a product that is in high demand in the market. Each product is sold at a price of $60. The cost invested for producing $x$ ten-thousand units is $(61x + \frac{100}{x} - 75)$ ten-thousand dollars $(1 \leqslant x \leqslant 50, x \in \mathbb{N}^*)$.

(I) Derive the analytical expression for the annual profit $y$ (in ten-thousand dollars) in terms of the annual production $x$ (in ten-thousand units).

(II) Using the property of the function $f(x) = x + \frac{a}{x}$ (where $a$ is a positive constant), which is decreasing on $(0, \sqrt{a}]$ and increasing on $[\sqrt{a}, +\infty)$, find the annual production that maximizes the profit in this product. (Annual profit = Total sales - Cost)

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