
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 guide students to "love reading, read more, and read good books," Class 7(2) of a certain school decided to purchase two types of books, $A$ and $B$. If they buy 1 book of type $A$ and 3 books of type $B$, it will cost a total of $180$ yuan; if they buy 3 books of type $A$ and 1 book of type $B$, it will cost a total of $140$ yuan.  
$(1)$ Find the cost of each book of type $A$ and type $B$.  
$(2)$ Based on the actual situation, the class requires that the total cost of purchasing books of type $A$ and $B$ does not exceed $700$ yuan, and the number of books of type $B$ is $\frac{3}{2}$ times the number of books of type $A$. Find all possible scenarios for the class to purchase books of type $A$ and $B.

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