
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 fruit with a cost of $10$ yuan per kilogram. It is found during the sales process that there is a linear relationship between the daily sales volume $y$ in kilograms and the selling price per kilogram $x$ yuan (where $10\leqslant x\leqslant 15$, and $x$ is an integer). When the selling price is $12$ yuan per kilogram, the daily sales volume is $90$ kilograms; when the selling price is $14$ yuan per kilogram, the daily sales volume is $80$ kilograms. <br/>$(1)$ Find the functional relationship between $y$ and $x$; <br/>$(2)$ If the supermarket wants to make a profit of $320$ yuan, what should be the selling price per kilogram? <br/>$(3)$ When the selling price per kilogram is set at what value, the supermarket's daily profit from selling this fruit is maximized, and what is the maximum profit?

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