
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 has a total of $10$ machines that produce a type of instrument component. Due to limitations in production capacity and technical level, a certain number of defective products ($p$) will be produced. According to experience, the relationship between the number of defective products per machine ($p$, in ten thousand pieces) and the daily output of each machine ($x$, in ten thousand pieces, $4 \leq x \leq 12$) satisfies the formula: $p=0.1x^{2}-3.2\ln x+3$. It is known that for each production of $10,000$ qualified components, a profit of $20,000$ can be made, but for each production of $10,000$ defective components, a loss of $10,000$ will be incurred.
(I) Express the daily profit ($y$, in ten thousand yuan) obtained by the factory in producing this type of component as a function of $x$.
(II) When the daily output ($x$, in ten thousand pieces) of each machine is optimal, what is the maximum profit?

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