
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 shopping mall is preparing to purchase two types of products, A and B, for sale. If the cost price of each product A is $2 less than the cost price of each product B, and the quantity of product A purchased with $80 is the same as the quantity of product B purchased with $100.  
$(1)$ Find the cost prices of each product A and B, respectively.  
$(2)$ If the quantity of product A purchased by the shopping mall is 5 fewer than 3 times the quantity of product B purchased, and the total quantity of products A and B purchased does not exceed 95, then how many units of product B can the shopping mall purchase at most?  
$(3)$ Under the conditions in $(2)$, if the selling prices of products A and B are $12 per unit and $15 per unit, respectively, and after selling all the purchased products A and B, the total profit from selling the two types of products exceeds $380, then what are the possible scenarios for the shopping mall to purchase products A and B?

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