
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 supermarket sells a certain type of vegetable. It is predicted that starting from the day of stocking, on the $n$th day $(n\in \mathbb{N}^{*}$ and $n\leqslant 5)$, the daily sales of this vegetable (in units of $kg$) are $100-10|n-3|$. It is known that the purchase price of this vegetable is $3$ yuan/kg, the selling price is $5$ yuan/kg, and any remaining vegetables can be disposed of at a price of $2$ yuan/kg. If the supermarket purchases $x$ kg of this vegetable every day $(80\leqslant x\leqslant 100)$, the total profit from selling this vegetable in the first $5$ days is denoted as $f(x)$ yuan.<br/>$(1)$ Find the analytical expression for $f(x)$;<br/>$(2)$ If the total sales volume of this vegetable sold at a price of $5$ yuan/kg in the first $5$ days, compared to the total purchase volume, is denoted as $Q$, and $g(x)=f(x)(1-Q)(80\leqslant x\leqslant 90)$, find the maximum and minimum values of $g(x)$.

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