
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
To create a bookish campus, I plan to purchase two types of bookshelves, A and B, to place the newly purchased books. A survey found that if 3 A bookshelves and 2 B bookshelves are purchased, a total of $1020$ yuan is needed; if 4 A bookshelves and 3 B bookshelves are purchased, a total of $1440$ yuan is needed.  
$(1)$ What is the price of each A and B bookshelf?  
$(2)$ If our school plans to purchase a total of $20$ bookshelves of these two types, with the number of B bookshelves not less than the number of A bookshelves, and the school can provide a maximum of $4320$ yuan, please provide the most cost-effective solution and calculate the cost of this solution.

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