
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 school organized a summer camp during the summer vacation. The school contacted two travel agencies, both of which quoted a price of $200 per person. After negotiation, Travel Agency A's discount condition was: a 25% discount for all teachers and students; Travel Agency B's discount condition was: the fee for one teacher could be waived, and the rest of the teachers and students would receive a 20% discount.  
$(1)$ Write down the function relationship between the cost $y$ (in dollars) and the number of teachers and students $x$ (people) for both travel agencies.  
$(2)$ For how many teachers and students is Travel Agency A cheaper than Travel Agency B?

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