
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
Early spring is the peak season for influenza A virus. To take preventive measures, our school intends to purchase several bottles of a brand of disinfectant with a volume of $200ml$ and several bottles of another brand of disinfectant with a volume of $500ml$. It is known that purchasing $3$ bottles of the first brand and $2$ bottles of the second brand costs $80$, and purchasing $1$ bottle of the first brand and $4$ bottles of the second brand costs $110$. <br/>$(1)$ Find the price per bottle of each brand of disinfectant; <br/>$(2)$ If our school needs to purchase a total of $4000ml$ of both brands of disinfectant, how many bottles of each brand (both brands need to be purchased) are needed? Please provide all possible purchasing plans; <br/>$(3)$ If our school spends $2500$ yuan on purchasing both brands of disinfectant, and there are a total of $1000 students and teachers in the school, each needing to use $10ml$ of disinfectant per day on average, then how many days can this batch of disinfectant last?

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