
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 purchased a certain product at a cost of $8$ yuan per item. During the sales process, it was found that there exists a certain function relationship between the daily sales quantity $y$ (in items) and the selling price per item $x$ (in yuan) (where $8\leqslant x\leqslant 15$, and $x$ is an integer). When $x=8$, $y=110$; when $x=10$, $y=100$; when $x=12$, $y=90$; and so on. Let $w$ represent the daily profit obtained by the supermarket from selling this disinfectant product.
1. Determine the type of function relationship between $y$ and $x$, and find the functional expression of $y$ in terms of $x$.
2. If the store makes a daily profit of $480$ yuan from selling this product, what is the selling price per item?
3. For what selling price per item will the daily sales profit be maximized, and what is the maximum profit?

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