
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
Every year, April 23rd is World Book Day. A certain school plans to purchase two types of bookshelves, $A$ and $B$, to store books. Through market research, it is found that purchasing 3 units of type $A$ bookshelves and 2 units of type $B$ bookshelves requires a total of $1020$ yuan; while purchasing 1 unit of type $A$ bookshelf and 3 units of type $B$ bookshelves requires a total of $900$ yuan.<br/>$(1)$ What are the unit prices of the $A$ and $B$ bookshelves, respectively?<br/>$(2)$ If the school plans to purchase a total of $20$ bookshelves of these two types, and the school can invest at most $4350$ yuan to purchase the bookshelves, how many units of type $B$ bookshelves can be purchased at most?

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