
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 plans to purchase a certain number of books of type $A$ and type $B$ from a bookstore for a student reading activity. The unit price of type $A$ books is $5$ yuan more than the unit price of type $B$ books. The number of type $A$ books purchased for $1200$ yuan is the same as the number of type $B$ books purchased for $900$ yuan.  
$(1)$ What are the unit prices of type $A$ and type $B$ books?  
$(2)$ According to the actual situation of the school, they need to purchase a total of $300$ books of type $A$ and type $B$ at once. When purchasing more than $100$ books of type $A$ and type $B$ at once, type $A$ books are discounted by $10%$ (type $B$ books are sold at the original price). If the total cost of purchasing type $A$ and type $B$ books does not exceed $5100$ yuan, how many type $A$ books can be purchased at most?

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