
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
With the rise in popularity of science fiction films and TV shows, more and more high school students are enjoying reading science fiction books. A certain school library purchased two types of science fiction books, labeled as type A and type B. It is known that the cost price of each type A book is $1.5$ times the cost price of each type B book. The number of type A books that can be purchased with $540$ yuan is $3$ less than the number of type B books that can be purchased with $450$ yuan. <br/>$(1)$ Find the cost price per book for type A and type B books, respectively. <br/>$(2)$ A certain high school plans to purchase a total of $50$ books of type A and type B, with the number of type A books being at least $6$ more than the number of type B books. How should the books be purchased to minimize the total cost $W$, and what is the minimum cost?

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