
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 supermarket purchases and sells two types of insulated cups, type A and type B. It is known that the price of purchasing 4 type A insulated cups and 5 type B insulated cups is the same. Also, purchasing 3 type A insulated cups costs $154 more than purchasing 2 type B insulated cups.  
$(1)$ Find the cost price of each type of insulated cup.  
$(2)$ Based on market demand, the supermarket decides to purchase a total of 80 of these two types of insulated cups for sale. Type A insulated cups are sold for $160 each, and type B insulated cups are sold for $140 each. After selling all of these two types of insulated cups, the supermarket makes a profit of $4100. Find out how many of each type of insulated cup the supermarket purchased this time.

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