
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
After the "Belt and Road" strategic concept was proposed in China, a technology company decided to develop and produce a large electronic device to seize the opportunities brought by the "Belt and Road". The annual fixed cost of producing this device is 5 million yuan, and for each production of $x$ units, an additional cost $c(x)$ (in million yuan) is required. When the annual output is less than 80 units, $c(x) = \frac{1}{2}x^2 + 40x$ (in million yuan); when the annual output is not less than 80 units, $c(x) = 101x + \frac{8100}{x} - 2180$ (in million yuan). If the selling price of each device is 100 million yuan, and the market analysis shows that the electronic devices produced by the company can be sold out.

(1) Find the function relationship of the annual profit $y$ (in million yuan) with respect to the annual output $x$ (in units);

(2) At what annual output does the company achieve the maximum profit in the production of this electronic device?

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