
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 tech company, in response to China's "One Belt, One Road" strategy, decides to develop and produce a large electronic device. The annual fixed cost for producing this device is 500 hundred-thousands yuan. Additionally, the cost $c(x)$ (in hundred-thousands yuan) for producing $x$ devices is $c(x) = \frac{1}{2}x^2 + 40x$ when the annual production is less than 80 devices, and $c(x) = 101x + \frac{8100}{x} - 2180$ when the annual production is no less than 80 devices. Each device is sold for 100 hundred-thousands yuan, and all produced devices can be sold.

(I) Find the relationship between the annual profit $y$ (in hundred-thousands yuan) and the annual production $x$ (in devices).

(II) How many devices should the company produce annually to maximize its profit from this electronic device?

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