
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
During the International Kite Festival in 2023, a distributor is preparing to purchase a batch of kites. It is known that the number of type $A$ kites that can be purchased with $20,000$ yuan is twice the number of type $B$ kites that can be purchased with $8,000$ yuan. The cost price of one type $A$ kite is $20$ yuan more than the cost price of one type $B$ kite.  
$(1)$ Find the cost price of one type $A$ and one type $B$ kite, respectively.  
$(2)$ After a market survey, it was found that the selling price of a type $A$ kite is always half the sum of the selling price and the sales volume of type $A$ kites, which is always equal to $130$. The selling price of type $B$ kites is $120$ yuan each. The distributor plans to purchase a total of $300$ type $A$ and type $B$ kites, with $m$ type $A$ kites ($50 \leq m \leq 150$). If all the kites can be sold, find the maximum profit from selling this batch of kites and provide the purchasing plan at that time.

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