
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 implement the prevention and control of spring flu, a certain school needs to purchase a batch of thermometers and disinfectant. If they buy $4$ thermometers and $2$ barrels of disinfectant, it will cost a total of $400$ yuan; if they buy $2$ thermometers and $4$ barrels of disinfectant, it will cost a total of $320$ yuan.  
$(1)$ Find the unit price of the thermometers and disinfectant.  
$(2)$ The school plans to purchase a total of $80$ items of these two types of supplies, and requires the number of thermometers to be no less than $\frac{1}{4}$ of the number of disinfectant. Design the most cost-effective purchasing plan, calculate the minimum cost, and explain the reason.

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