
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
During the fight against the epidemic, a certain store purchased a type of disinfectant product at a cost of $8$ yuan per item. It was found during the sales process that there is a linear relationship between the daily sales quantity $y$ (items) and the selling price per item $x$ (yuan) (where $8\leqslant x\leqslant 15$, and $x$ is an integer). Some corresponding values are shown in the table below:

| Selling Price (yuan) | $9$ | $11$ | $13$ |
|----------------------|-----|------|------|
| Daily Sales Quantity (items) | $105$ | $95$ | $85$ |

$(1)$ Find the function relationship between $y$ and $x$.  
$(2)$ If the store makes a profit of $425$ yuan per day selling this disinfectant product, what is the selling price per item?  
$(3)$ Let the store's profit from selling this disinfectant product per day be $w$ (yuan). When the selling price per item is what amount, the daily sales profit is maximized? What is the maximum profit?

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