
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 enhance students' interest in reading, a certain school has set up a shared bookshelf and purchased a batch of books. The school spent $3000$ yuan on purchasing $A$ types of books and $1600$ yuan on purchasing $B$ types of books. The unit price of $A$ types of books is $1.5$ times that of $B$ types of books, and the number of $A$ types of books purchased is $20$ more than the number of $B$ types of books purchased.  
$(1)$ Find the unit prices of the two types of books $A$ and $B$;  
$(2)$ The bookstore held a discount promotion on "World Book Day," selling all books at an $20\%$ discount. The school purchased $20$ books of type $A$ and $25$ books of type $B$ on that day. How much did they spend in total?

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