
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
"Aibei pickled mustard" is a famous specialty of Huicheng District. A supermarket sells two brands of pickled mustard, $A$ and $B$. If you buy 9 packs of brand $A$ pickled mustard and 6 packs of brand $B$ pickled mustard, the total cost is $390$ yuan; if you buy 5 packs of brand $A$ pickled mustard and 8 packs of brand $B$ pickled mustard, the total cost is $310$ yuan. <br/>$(1)$ What is the price per pack of brand $A$ pickled mustard and brand $B$ pickled mustard, respectively? <br/>$(2)$ If a company buys a total of 30 packs of brand $A$ and brand $B$ pickled mustard, with the quantity of brand $A$ being at least 5 packs 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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