
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 student supplies supermarket is preparing to purchase two types of pencil cases, $A$ and $B$, for sale. If the cost price of each $A$ type pencil case is $2$ yuan less than that of each $B$ type pencil case, and the quantity of $A$ type pencil cases purchased with $800$ yuan is the same as the quantity of $B$ type pencil cases purchased with $1000$ yuan.  
$(1)$ What are the cost prices of each $A$ type and $B$ type pencil cases, respectively?  
$(2)$ Let the supermarket purchase $m$ $B$ type pencil cases.  
① If the quantity of $A$ type pencil cases purchased is $50$ less than three times the quantity of $B$ type pencil cases purchased, and the total quantity of $A$ and $B$ type pencil cases purchased does not exceed $910$, how many $B$ type pencil cases can the supermarket purchase at most?  
② Under the conditions of ①, if the selling prices of $A$ type and $B$ type pencil cases are $12$ yuan per case and $15$ yuan per case, respectively, and after selling all the purchased $A$ and $B$ type pencil cases, the total profit from selling the two types of pencil cases exceeds $3795$, then how many different scenarios are there for the supermarket to purchase the two types of pencil cases?

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