
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
Xinhua Bookstore is preparing to order a batch of books. There are two suppliers, A and B, both priced at $40$ yuan per book. To promote sales, A said: "All books purchased at my place will be discounted by 10%." B said: "If you purchase more than $100$ books, the excess will be discounted by 20%."  
$(1)$ If Xinhua Bookstore is preparing to order $150$ books, please calculate the amount of money that needs to be paid at both A and B.  
$(2)$ If the amount of money that Xinhua Bookstore needs to pay at both A and B is the same, then how many books is Xinhua Bookstore preparing to order?

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