
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
In the 2023 Hangzhou Asian Games, the mascot was very popular as soon as it went on sale. A certain tourist shop purchased a certain type of Asian Games mascot at a price of $50$ yuan per item and sold it at a price of $80$ yuan per item. They were able to sell $200$ items per day. Starting from July, the store decided to use price reductions to promote sales. Through experiments, it was found that for every $1$ yuan reduction in price, the daily sales volume would increase by $20$ items.  
$(1)$ Let the set price be $x$ yuan and the daily sales volume be $y$ items. Express $y$ in terms of $x$, $y=$______;  
$(2)$ At what price should the mascot be sold so that the daily sales profit reaches $7500$ yuan?  
$(3)$ Calculate the pricing strategy that would maximize the daily sales profit.

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