
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
Vaccination is an effective way to prevent the spread of the novel coronavirus. There are two community vaccination points, A and B. It is known that the average number of people vaccinated per day at point A is $1.25$ times the average number of people vaccinated per day at point B. In addition, the time required for point A to complete the vaccination of 3000 people is 2 days less than the time required for point B to complete the vaccination of 4000 people.  
$(1)$ Find the average number of people vaccinated per day at points A and B.  
$(2)$ After a period of time, point B increased its publicity efforts. The average number of people vaccinated per day at this point increased by $25\%$ compared to the original average. Due to the influence of the publicity at point B, the average number of people vaccinated per day at point A decreased by $5m$ people, but not less than 800 people. In this case, the number of people vaccinated at point B for $\left(m+15\right)$ days is $6000$ more than the number of people vaccinated at point A for $2m$ days. Find the value of $m$.

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