
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
To effectively prevent the spread of infectious diseases, a school needs to purchase two types of disinfectants, A and B, to disinfect the classrooms every day. It is known that the selling price of disinfectant A is $10$ yuan more than that of disinfectant B. The school purchased the same number of barrels of disinfectant A and B for $600$ yuan and $400$ yuan, respectively.<br/>$(1)$ Find the selling price of disinfectants A and B per barrel, respectively;<br/>$(2)$ Due to the need for disinfection work, the school needs to purchase a total of $500$ barrels of the two disinfectants, with the number of barrels of disinfectant A not less than the number of barrels of disinfectant B. Find the number of barrels of disinfectant A to be purchased when the total amount of funds required is minimized, and what is the minimum total amount?

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