
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]


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A certain ancient town, in order to develop the tourism industry, attract more tourists to visit, and help rural revitalization, decided to offer a special ticket discount for group tours during the "May Day" period, with prices as shown in the table below:

| Number of Ticket Buyers $m$ (people) | $10\leqslant m\leqslant 50$ | $51\leqslant m\leqslant 100$ | $m \gt 100$ |
|-------------------------------------|-----------------------------|------------------------------|-------------|
| Price per person (yuan)             | $60$                        | $50$                         | $40$        |

*Note: The number of people in the group in the question is not less than 10.*

Currently, there are two groups, A and B, with a total of 102 people, planning to travel to the ancient town during the "May Day" holiday. Group A has fewer than 50 people, and Group B has more than 50 people.

$(1)$ If the two groups buy tickets separately, the total amount to be paid is 5580 yuan. How many people are in Groups A and B respectively?

$(2)$ If the two groups join together to purchase tickets as a "large group," saving at least 1200 yuan compared to each group buying tickets separately, what is the minimum number of people in Group A?

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