
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 in our city sells a type of stationery. The cost price is $5$ yuan per item. When the selling price is $6$ yuan per item, the sales volume for that day is $100$ items. It was found during the sales process that for every increase of $0.5$ yuan in the selling price, the sales volume for that day decreases by $5$ items. Let the uniform selling price for that day be $x$ yuan per item $(x\geqslant 6$, and $x$ increases by multiples of $0.5$ yuan). The profit from sales for that day is $y$ yuan.
$(1)$ Find the functional relationship between $y$ and $x$ (the range of the independent variable values is not required);
$(2)$ In order to reduce inventory as soon as possible, what should be the selling price per item for that day if the profit from sales for that day should be $240$ yuan?

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