
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 store sells regular exercise books and deluxe exercise books. The total sales amount for $150$ regular exercise books and $100$ deluxe exercise books is $1450$ yuan; the total sales amount for $200$ regular exercise books and $50$ deluxe exercise books is $1100$ yuan.  
$(1)$ Find the unit prices of regular exercise books and deluxe exercise books.  
$(2)$ The store plans to purchase another $500$ exercise books, with the quantity of regular exercise books not less than three times the quantity of deluxe exercise books. It is known that the cost price of regular exercise books is $2$ yuan per book, and the cost price of deluxe exercise books is $7$ yuan per book. Let $x$ be the number of regular exercise books purchased, and $W$ be the profit obtained.  
① Find the functional relationship of $W$ with respect to $x$.  
② How should the store purchase the goods to maximize the total profit? Find the maximum profit.

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