
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
The school plans to organize 600 teachers and students from the seventh grade to rent a car for a study tour. It is known that there are two types of buses in the rental company, one with 32 seats and the other with 45 seats. The total rent for one 32-seat bus and two 45-seat buses is 2800 yuan. The rent for each 45-seat bus is exactly 1.25 times the rent for each 32-seat bus.  
$(1)$ Find the rent for each 32-seat bus and each 45-seat bus.  
$(2)$ If one of these two types of vehicles is rented separately (all teachers and students can ride, and only one vehicle may not be full), how much is needed for each?  
$(3)$ If the school rents a total of 14 buses of these two types at the same time (all teachers and students can ride, and only one vehicle may not be full), write directly the number of each type of bus when the rental cost is the least, and find the rental cost at this time.

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