
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 is organizing a visit to a museum for students, with each ticket costing $30$ yuan. It is known that there are two discount options for group tickets. Option one: a 30% discount for all members; option two: an 20% discount for all members and 5 people can enter for free.<br/>$(1)$ A group of 45 students, which option is more cost-effective?<br/>$(2)$ Regardless of the option chosen, the cost of purchasing tickets for a second group is the same. Find the number of students in the second group. (Solve using a system of linear equations)

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