
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 a field trip for seventh-grade students. If several $35$-seat buses are rented individually, they will be filled exactly; if $45$-seat buses are rented individually, one less bus is needed, with $25$ extra seats remaining.<br/>$(1)$ Find the number of students participating in the field trip for seventh-grade students at this school;<br/>$(2)$ It is known that the rental fee for a $35$-seat bus is $320$ yuan per bus, and for a $45$-seat bus, it is $380$ yuan per bus. If the school decides to rent a total of $6$ buses of both types simultaneously (not necessarily filled to capacity), please help the school design the most cost-effective rental plan.

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