
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
On a purchase order from a certain mobile phone specialty store, the following information is provided: The unit price of type $A$ phones is $800$ yuan more than that of type $B$ phones. The quantity of type $A$ phones purchased for $38400$ yuan is the same as the quantity of type $B$ phones purchased for $28800$ yuan.  
$(1)$ Find the unit prices of type $A$ and type $B$ phones.  
$(2)$ If the selling price of type $A$ phones is $3700$ yuan and the selling price of type $B$ phones is $2700$ yuan. The store wants to spend $28000$ yuan to purchase several units of type $A$ and type $B$ phones. What are the possible purchase plans? Calculate which plan will yield the highest total profit.

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