
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 certain high-tech enterprise plans to build a new production line to produce new equipment, in order to accelerate the development of the enterprise. It is known that the annual fixed cost of this production line is $5$ million yuan. If $x$ ($x\in N*$) new equipment is produced each year, an additional variable cost of $y_{1}$ million yuan is required. When the annual output $x$ is less than $80$ units, $y_1=\frac{1}{2}x^2+40x$; when the annual output $x$ is greater than or equal to $80$ units, $y_1=101x+\frac{8100}{x}-2180$. The selling price of each new equipment is $100$ million yuan. Through market analysis, it is known that all the new equipment produced by the enterprise can be sold in the same year. Let $f(x)$ million yuan be the annual profit generated by this production line when the annual output is $x$ units.<br/>$(1)$ Write down the analytical expression of $f(x)$;<br/>$(2)$ For how many units of annual output $x$ does the production line achieve the maximum annual profit?

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