
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 is preparing to purchase two types of eye-protection lamps, $A$ and $B$. It is known that the cost price of each lamp of type $A$ is $40$ yuan more than that of type $B$. The quantity of type $A$ lamps that can be purchased with $2000$ yuan is the same as the quantity of type $B$ lamps that can be purchased with $1600$ yuan.  
$(1)$ What is the cost price of each lamp of type $A$ and type $B$?  
$(2)$ The store plans to purchase a total of $80$ lamps of types $A$ and $B$ with a total budget not exceeding $14550$ yuan. The selling prices of type $A and type $B lamps are $300$ yuan and $200$ yuan per lamp, respectively. If all the lamps are sold, how should the store purchase the lamps to maximize profit? What is the maximum profit?

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