
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 school organized students to go on a study tour. The travel agency charges $300$ yuan per person. When the number of study tour participants exceeds $50$, the travel agency offers two discount schemes:<br/>Scheme 1: The study group pays $1500$ yuan upfront, and then each person is charged $240$ yuan;<br/>Scheme 2: $5$ people are free, and the rest are charged at a 10% discount (10% off the original price).<br/>$(1)$ Express in algebraic terms, when the total number of study tour participants is $x\left(x \gt 50\right)$,<br/>the total charge under Scheme 1 is ______ yuan;<br/>the total charge under Scheme 2 is ______ yuan;<br/>$(2)$ When the total number of participants in the tour is $80$ people, which scheme saves money? Explain your reasoning.

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