
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
The school library purchased a batch of books and found that two sets of books were particularly suitable for student reading. One set is a series of revolutionary stories consisting of 18 books, and the other set is a series of popular science books for teenagers consisting of 20 books. If purchasing 3 sets of revolutionary story series and 2 sets of popular science books for teenagers costs a total of 1290 yuan, and purchasing 2 sets of revolutionary story series and 3 sets of popular science books for teenagers costs a total of 1260 yuan;  
$(1)$ If the school budget is 500 yuan, can they each buy a set of books?  
$(2)$ After the school buys one set of each book, they find that these two sets of books are particularly popular among students and decide to purchase them again. They plan to buy the single book from the revolutionary story series that students like the most (each book has the same price) and another set of popular science books for teenagers. At this point, the bookstore is having a promotion where the full set of popular science books for teenagers is discounted by 10%, and the price of each book in the revolutionary story series will increase by 20%. If the school budget is 300 yuan, how many books from the revolutionary story series can they buy at most?

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