
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 school organized students to study at the Jinggangshan Revolution Museum. The travel agency charges $400$ yuan per person. When the number of study participants exceeds $50$, the travel agency offers two discount options:<br/>Option 1: The study group pays $1500$ yuan upfront, and then each person is charged $320$ yuan.<br/>Option 2: $5$ people are free, and the rest are charged at a 10% discount.<br/>$(1)$ When the total number of study participants is $x\left(x \gt 50\right)$, please express with an expression containing $x$:<br/>① The total charge for Option 1 is ______ yuan.<br/>② The total charge for Option 2 is ______ yuan.<br/>$(2)$ When the total number of travelers is $80$, which option saves money? Please determine and explain the reason.

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