
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
A certain school wants to purchase two types of disinfectants, A and B, to prevent the novel coronavirus. If they buy 9 barrels of disinfectant A and 6 barrels of disinfectant B, the total cost is $615$ yuan; if they buy 8 barrels of disinfectant A and 12 barrels of disinfectant B, the total cost is $780$ yuan.  
$(1)$ What is the price per barrel of disinfectant A and disinfectant B, respectively?  
$(2)$ If the school plans to purchase a total of 30 barrels of disinfectant A and B, with $a$ barrels of disinfectant A, and the quantity of disinfectant A is at least 5 barrels more than the quantity of disinfectant B, but not more than twice the quantity of disinfectant B. How should they purchase to minimize the total cost $W$? And what is the minimum cost?

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