
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 goods, A and B, for sale. If the cost price of each type A item is $2 less than the cost price of each type B item, and the quantity of type A items that can be purchased with $80 is the same as the quantity of type B items that can be purchased with $100.  
$(1)$ Find the cost price of each type A and type B item.  
$(2)$ If the shopping mall purchases 5 fewer type A items than 3 times the quantity of type B items, and the selling prices of type A and type B items are $12 per item and $15 per item, respectively, and after selling all the purchased type A and type B items, the total profit exceeds $380, then how many type B items should the shopping mall purchase at least?

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