
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
In a certain junior high school, the 7th grade is preparing to watch the movie "Volunteers". Each class monitor is responsible for buying tickets, and the number of students in each class is more than 40. The ticket price is $30$ yuan each. A class monitor asked the ticket seller if group tickets could be discounted. The ticket seller said that there are two discount options for groups of more than 40 people: <br/> Option 1: All members can get a $20\%$ discount; Option 2: If a $10\%$ discount is given, 6 people can get free tickets. <br/> $(1)$ If the second class has 50 students, which option should they choose? <br/> $(2)$ After thinking for a while, a class monitor from the first class said, "No matter which option we choose, the amount of money we have to pay is the same. Do you know how many students are in the first class?

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