
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 Dragon Boat Festival is a traditional festival in China that has been selected as a world intangible cultural heritage. Eating zongzi during the Dragon Boat Festival is a traditional custom of the Chinese nation. In order to meet people's needs, a supermarket plans to purchase two types of zongzi, A and B, for sale before the Dragon Boat Festival. It is understood that the cost price of each type B zongzi is $2 more than the cost price of each type A zongzi. The number of type A zongzi that can be purchased for $1000 is the same as the number of type B zongzi that can be purchased for $1200.  
$(1)$ What is the cost price of each type A and type B zongzi?  
$(2)$ The supermarket plans to purchase a total of 200 zongzi (both types), with the number of type A zongzi being at least twice the number of type B zongzi. If the selling prices of type A and type B zongzi are $12/$ each and $15/$ each, respectively, and the profit obtained when all zongzi are sold out is $W.  
① Find the functional relationship between $W$ and $m$, and determine the range of values for $m$;  
② How should the supermarket purchase the goods to maximize profit, and what is the maximum profit?

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