
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 Chinese festival where people have the custom of eating zongzi. As the Dragon Boat Festival approaches this year, a certain shopping mall predicts that type $A$ zongzi will sell well. According to the prediction, the cost price of type $A$ zongzi per kilogram before the festival is $2$ yuan more than after the festival. The quantity of type $A$ zongzi purchased for $240$ yuan before the festival is $4$ kilograms less than the quantity purchased for the same amount after the festival. Based on the above information, answer the following questions:  
$(1)$ What is the cost price per kilogram of type $A$ zongzi after the festival at this shopping mall?  
$(2)$ If the shopping mall purchases a total of $400$ kilograms of type $A$ zongzi before and after the festival, and the total cost does not exceed $4600$ yuan, and all zongzi are sold at $20$ yuan per kilogram before the festival and $16$ yuan per kilogram after the festival, how many kilograms of type $A$ zongzi should the shopping mall purchase before the festival to maximize profit? What is the maximum profit?

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