
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
"Guang'an Salted Duck Eggs" is a famous specialty of Xiaoping's hometown. A certain supermarket sells two brands of salted duck eggs, $A$ and $B$. If you buy 9 boxes of brand $A$ salted duck eggs and 6 boxes of brand $B$ salted duck eggs, the total cost is $390$ yuan; if you buy 5 boxes of brand $A$ salted duck eggs and 8 boxes of brand $B$ salted duck eggs, the total cost is $310$ yuan.  
$(1)$ What is the price per box of brand $A$ and brand $B$ salted duck eggs, respectively?  
$(2)$ If a company buys a total of $30$ boxes of brand $A$ and brand $B$ salted duck eggs, with the quantity of brand $A$ being at least 5 boxes more than brand $B$ but not exceeding twice the quantity of brand $B$, how should they purchase to minimize the total cost? And what is the minimum cost?

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