
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" initiative was proposed in China, a technology company decided to develop and produce a large electronic device to seize the opportunities brought by the initiative. The annual fixed cost of producing this device is $5$ million yuan, and an additional cost of $c(x)$ million yuan is required for producing $x$ units. When the annual output is less than $90$ units, $c(x)=\frac{1}{2}x^{2}+60x$ million yuan; when the annual output is not less than $90$ units, $c(x)=121x+\frac{8100}{x}-2180$ million yuan. If the selling price of each device is $1.2$ million yuan, through market analysis, the company can sell all the electronic devices produced.<br/>$(1)$ Find the function relationship of annual profit $y$ (in million yuan) with annual output $x$ (in units);<br/>$(2)$ At what annual output does the company maximize profit in the production of this electronic device?

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