
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
Labor Day is approaching. Travel agencies A and B have proposed group discount methods to attract more customers to travel to a certain destination. The discount method of travel agency A is: buy 4 full-price tickets, and the rest of the people will receive a half-price discount; the discount method of travel agency B is: all customers will receive a 30% discount. It is known that the original price of both travel agencies is $200 per person.
$(1)$ Express the relationship between the charges of travel agency A, $y_{1}$, the charges of travel agency B, $y_{2}$, and the number of travelers, $x$.
$(2)$ Discuss which travel agency offers a more favorable charge based on the number of people participating in the trip.

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