
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]


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After the strategic concept of China's "Belt and Road" was proposed, a technology company decided to seize the opportunities brought by the "Belt and Road" by developing and producing a large electronic device. The annual fixed cost of producing this device is 5 million yuan, and the additional cost $C(x)$ (in million yuan) for producing $x$ units is as follows: 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 $x$ (units);

$(2)$ At what annual output will 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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