
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 supermarket sells a "no-wash hand sanitizer" with a cost price of $16$ yuan per bottle. When the selling price is set at $20$ yuan, they can sell $80$ bottles per day. According to market research, it is decided to reduce the price for sales. The market survey reflects that for every decrease of $0.5$ yuan in the selling price, they can sell an additional $20$ bottles per day (the selling price is not lower than the cost price). If the selling price of this "no-wash hand sanitizer" is denoted as $x$ (yuan) and the daily sales volume is denoted as $y$ (bottles), find:<br/>$(1)$ the functional relationship between the daily sales volume $y$ (bottles) and the selling price $x$ (yuan);<br/>$(2)$ at what price per unit should the "no-wash hand sanitizer" be sold for the daily sales profit to be maximized, and what is the maximum profit in yuan?

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