
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
To broaden students' horizons, a certain school organized a study tour to the "Eighth Wonder of the World" - the Terracotta Warriors and Horses. In this activity, students like Xiao Liang and Xiao Hong went to the scenic spot with their teacher. It is known that the adult ticket is $120$ yuan each, and the student ticket is discounted by 50% compared to the adult ticket. There were a total of $130$ people, and the total cost of tickets was $9600$ yuan.  
$(1)$ How many adults and how many students went on the trip?  
$(2)$ If the group ticket (50 people or more) is discounted by 40% per person compared to the adult ticket, please help Xiao Liang calculate how to save money on ticket purchases.

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