
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
"群防群控,众志成城,遏制疫情,我们一定能赢!" In order to prepare for the reopening of school, a certain school purchased a total of 30 barrels of two types of barrel-packaged disinfectants, A and B, for campus disinfection. It is known that type A disinfectant costs 300 yuan per barrel and can cover an area of 2000 square meters for disinfection, while type B disinfectant costs 200 yuan per barrel and can cover an area of 1000 square meters for disinfection.  
$(1)$ Let $x$ represent the number of barrels of type A disinfectant purchased, and $y$ represent the cost of purchasing disinfectant. Write the relationship between $y$ and $x$.  
$(2)$ Under the condition that the current funds do not exceed 8200 yuan, find the maximum area that can be disinfected.

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