
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 online store sells a type of children's clothing. The cost price of each item is $50$ yuan, and the current selling price is $80$ yuan per item. The store can sell 200 items per week. During the Double Twelve promotion period, the store decides to reduce the price to promote sales. After market research, it is found that for every $1$ yuan reduction in price, an additional 20 items can be sold per week.  
$(1)$ If the store wants to achieve a weekly profit of $7500$ yuan while maximizing customer benefits, what should be the selling price per item?  
$(2)$ What should be the selling price per item for the store to maximize its weekly profit? What is the maximum weekly profit?

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