
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 store purchased a type of disinfectant product at a cost of $8$ yuan per item. During the sales process, it was found 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). When the selling price per item of the disinfectant product is $9$ yuan, the daily sales quantity is $105$ items; when the selling price per item is $11$ yuan, the daily sales quantity is $95$ items. <br/>$(1)$ Find the functional relationship between $y$ and $x$; <br/>$(2)$ Suppose the store makes a profit of $w$ yuan per day selling this disinfectant product. At what price per item should the disinfectant product be sold to maximize the daily profit? What is the maximum profit?

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