
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
A certain car beauty company launched a promotional activity to attract customers: for first-time customers, the charge is $200 per visit, and they can register as members. Members will receive corresponding discounts for each subsequent visit, as shown in the table below:

| Number of Visits | 1st | 2nd | 3rd | 4th | 5th and above |
|------------------|-----|-----|-----|-----|---------------|
| Charge Ratio     | 1   | 0.95| 0.90| 0.85| 0.80          |

The company randomly selected 100 registered members for a survey and obtained the following statistical data:

| Number of Visits | 1st | 2nd | 3rd | 4th | 5th and above |
|------------------|-----|-----|-----|-----|---------------|
| Frequency        | 60  | 20  | 10  | 5   | 5             |

Assuming the cost for the car beauty service is $150 each time, answer the following questions based on the given data:

$(1)$ Estimate the probability that a member of the company will consume at least twice.

$(2)$ If a member only consumes twice, calculate the average profit the company will make from these two visits.

$(3)$ The company wants to randomly sample 6 customers who have visited twice and three times from the 100 members using stratified random sampling. Then, 2 customers will be selected from these 6 customers to receive souvenirs. Calculate the probability that exactly 1 of the 2 selected customers has visited twice.

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