
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
It is known that the cost price of each book of type A is $20$ yuan less than the cost price of each book of type B. The quantity of type A books purchased for $540$ yuan is the same as the quantity of type B books purchased for $780$ yuan.  
$(1)$ Find the cost price of each book of type A and type B.  
$(2)$ A certain middle school plans to purchase a total of $70$ books of type A and type B, with the total cost not exceeding $3550$ yuan. How many books of type A should be purchased at least?

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