
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 middle school plans to hold a knowledge competition with the theme "Strive for a hundred years, set sail for a new journey," and rewards the winning students. They now want to purchase two types of prizes, A and B. It is known that 1 piece of prize A and 2 pieces of prize B together cost $40, and 2 pieces of prize A and 3 pieces of prize B together cost $70.  
$(1)$ Find the unit prices of prizes A and B.  
$(2)$ According to the award plan, the school needs a total of 60 pieces of prizes A and B, with at least 20 pieces of prize A. How should they purchase to minimize the total cost? And what is the minimum cost?

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