
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 sports store purchases two types of soccer balls, $A$ and $B$, as the new school year approaches. It is known that the cost price of each $A$ type soccer ball is $40$ yuan higher than the cost price of each $B$ type soccer ball. The quantity of $A$ type soccer balls purchased for $480$ yuan is equal to the quantity of $B$ type soccer balls purchased for $240$ yuan.  
$(1)$ Find the cost price of each $A$ type soccer ball and each $B$ type soccer ball.  
$(2)$ The store plans to purchase a batch of $A$ and $B$ type soccer balls for $4000$ yuan. The selling price of each $A$ type soccer ball is $100$ yuan, and the selling price of each $B$ type soccer ball is $55$ yuan. Let the total profit be $W$ yuan. If the store wants to purchase no fewer $A$ type soccer balls than $B$ type soccer balls, how should they purchase the balls to maximize the total profit $W$?

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