
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
The Jincheng Greenway in Chengdu is a first-class greenway in the newly opened circular ecological park. The beautiful scenery attracts many citizens to buy bicycles for riding. A certain bicycle shop plans to purchase two models, $A$ and $B$, of road bikes totaling 50 bikes. Each $B$ model road bike costs $600 more than each $A$ model road bike. The number of $A$ model road bikes purchased for $5000 is the same as the number of $B$ model road bikes purchased for $8000. <br/>$(1)$ Find the purchase price of the two models of road bikes, $A$ and $B$; <br/>$(2)$ If the store plans to purchase $m$ $A$ model road bikes, invest a maximum of $68000, and the number of $B$ model road bikes cannot be less than the number of $A$ model road bikes, what are the possible purchase plans for the store? <br/>$(3)$ Under the conditions of $(2)$, if the selling price of each $A$ model road bike is $1500, and each $B$ model road bike is $2000, how should the store purchase to maximize profit? What is the maximum profit at this time?

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