
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
Recently, schools in Beijing, Shanghai, Zhejiang, Tianjin, and other places have suspended classes due to students contracting influenza A. Influenza A refers to influenza caused by the influenza A virus, an acute respiratory infectious disease. In order to prevent the spread of the influenza A virus, schools are preparing to purchase a batch of medical masks and hand sanitizers for daily protection. If 500 medical masks and 40 bottles of hand sanitizer are purchased, it will cost 1250 yuan; if 1000 medical masks and 20 bottles of hand sanitizer are purchased, it will cost 1000 yuan.  
$(1)$ Find the unit prices of medical masks and hand sanitizer.  
$(2)$ The school has prepared 400 yuan for this purchase. In addition to purchasing medical masks and hand sanitizer, they also need to increase the purchase of N95 masks priced at 2 yuan each by $a$ units. The total number of medical masks and N95 masks is 200. They also need to buy $b$ bottles of hand sanitizer. All the money is spent exactly, how many purchasing plans does the school have in total? Write down all the purchasing plans.

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