
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
The famous school bands A and B in Wenzhou city decided to purchase the same performance costumes from a clothing factory. The table below shows the price list of the performance costumes provided by the clothing factory:

| Number of Sets Purchased | $1\sim 39$ sets (including 39 sets) | $40\sim 79$ sets (including 79 sets) | 80 sets and above |
|--------------------------|-------------------------------------|--------------------------------------|-------------------|
| Price per Set            | $80$ yuan                           | $70$ yuan                            | $60$ yuan         |

After investigation, it was found that the two bands have a total of 75 members (Band A has no fewer than 40 members). If they each purchase their own performance costumes, the total cost for both bands would be $5600$ yuan. Please answer the following questions:

$(1)$ If bands A and B join together to purchase the costumes, how much money can they save at most compared to purchasing separately?

$(2)$ How many students are there in bands A and B respectively?

$(3)$ Now, $a$ members are selected from band A and $b$ members are selected from band B (with each band having no fewer than 5 members) to perform at a charity event at a children's welfare home. After the performance, each band member will engage in a "heart-to-heart activity" with the children; each member of band A will be responsible for 3 children, and each member of band B will be responsible for 5 children. This arrangement ensures that all 65 children at the welfare home receive the warmth of the "heart-to-heart activity." Please list all possible selection plans and explain the reasons.

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