
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 the first traditional Chinese festival to be listed as a UNESCO Intangible Cultural Heritage. It falls on the fifth day of the fifth month of the lunar calendar each year. There are customs such as dragon boat racing and eating zongzi. A certain shopping mall is preparing to purchase and sell two types of zongzi, $A$ and $B$, in anticipation of the Dragon Boat Festival. It is known that the quantity of zongzi $A$ that can be purchased for $3000$ yuan is $40$ more than the quantity of zongzi $B$ that can be purchased for $3360$ yuan, and the unit price of zongzi $B$ is $1.2$ times the unit price of zongzi $A$. <br/>$(1)$ Find the unit prices of zongzi $A$ and zongzi $B$. <br/>$(2)$ If the shopping mall plans to purchase a total of $2200$ zongzi for sale, and the cost of purchasing zongzi $A$ is not more than the cost of purchasing zongzi $B$, write down the relationship between the total cost $y$ (in yuan) and the quantity $a$ (in pieces) of zongzi $A$, and determine how to make the total cost the lowest. What is the minimum cost?

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