
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
May 13th is Mother's Day. In order to welcome Mother's Day, the Likeman Department Store plans to purchase a batch of two types of toys, type A and type B. It is known that the cost of one type A toy plus the cost of one type B toy is $40. The number of type A toys purchased with $90 is the same as the number of type B toys purchased with $150.
$(1)$ Find the cost of each type A and type B toy.
$(2)$ The store plans to purchase a total of 48 type A and type B toys, with the number of type A toys being less than 24. The store also decides that the total amount spent on this purchase will not exceed $1000. How many different purchasing plans does the store have?
$(3)$ Under the conditions in $(2)$, if the selling price of each type A toy is $30 and each type B toy is $45, find the relationship between the profit $W (in dollars)$ from selling all these toys and the quantity $m (in pieces)$ of type A toys purchased. Also, determine the maximum profit.

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