
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 hand sanitizer, which costs $16$ yuan per bottle. When the selling price is set at $20$ yuan per bottle, they can sell $60$ bottles per day. Market research shows that for every $1$ yuan increase in the selling price, they sell $5$ fewer bottles per day. If the selling price of this hand sanitizer increases by $x$ yuan, the daily profit from sales is $y$ yuan.  
$(1)$ The daily sales volume is ______ bottles, and the profit per bottle of hand sanitizer is ______ yuan. (Express in algebraic expressions containing $x$)  
$(2)$ If the daily profit $y$ from the sales of this hand sanitizer reaches $300$ yuan, how much should the selling price increase?  
$(3)$ When the selling price increases by how many yuan, the daily profit $y$ of this hand sanitizer is maximized, and what is the maximum profit in yuan?

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