
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
Tea is the national drink of China, and tea culture is an important part of traditional Chinese culture, which has also driven the extension and development of related cultures such as tea art, tea ware, and tea clothing. During the "Spring Tea Festival," a tea ware shop owner purchased two different types of tea sets, $A$ and $B$. If the owner buys 1 set of type $A$ and 2 sets of type $B$, it costs $250$ yuan. If the owner buys 3 sets of type $A$ and 4 sets of type $B$, it costs $600$ yuan. <br/>$(1)$ What is the cost price per set for type $A$ and type $B$ tea sets? <br/>$(2)$ Due to the high demand for tea sets, the owner decides to purchase a total of 80 sets of type $A$ and type $B$ tea sets. The tea set factory adjusted the prices of the two types of tea sets: the cost price of type $A$ tea sets increased by $8\%$ compared to the first purchase, and the cost price of type $B$ tea sets is 80% of the original cost price. If the total cost for purchasing type $A$ and type $B$ tea sets does not exceed $6240$ yuan, how many sets of type $A$ tea sets can the owner purchase at most? <br/>$(3)$ If a profit of $30$ yuan can be made by selling one set of type $A$ tea sets and a profit of $20$ yuan can be made by selling one set of type $B$ tea sets, under the conditions of $(2)$, how should the owner purchase the tea sets to maximize the profit? What is the maximum profit?

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