
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
During this year's May Day holiday, our city experienced a peak in short-term tourism. The ticket prices for a popular attraction are as follows:

| Type of Ticket | $A$ | $B$ | $C$ |
| --- | --- | --- | --- |
| Number of People per Ticket | $1\sim 50$ | $51\sim 100$ | $100$ and above |
| Price per Ticket (in yuan) | $50$ | $45$ | $40$ |

A travel agency received two tour groups, A and B, with a total of $102$ people (group A has more people than group B). When planning to purchase tickets, if the two groups purchase tickets jointly as one group, they would save $730$ yuan compared to purchasing tickets separately.

$(1)$ Find out how many people are in each of the two tour groups.

$(2)$ For a tour group with fewer than $50$ people, what is the minimum number of people for which purchasing type $B$ tickets instead of type $A$ tickets would result in savings?

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