
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 school is organizing a textbook drama display activity and plans to purchase two types of prizes, $A$ and $B$. It is known that purchasing 3 items of prize $A$ and 2 items of prize $B$ will cost a total of $390$ yuan. The cost of 4 items of prize $A$ is $60$ yuan more than the cost of 5 items of prize $B$.  
$(1)$ Find the cost of each item of prize $A$ and prize $B$.  
$(2)$ If the school plans to purchase a total of 30 items of prizes $A$ and $B$, and the quantity of prize $A$ is not less than half of the quantity of prize $B$, and the total cost of purchasing the two types of prizes does not exceed $2170$ yuan, how should the school make the purchases to minimize the total cost? What is the minimum cost?

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