
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 promotes students' comprehensive development in morality, intelligence, physical fitness, and aesthetics through labor education. The school plans to organize eighth-grade students to carry out labor practice activities at the "Happy" Farm. Upon arrival at the farm, students are divided into groups for labor. If each teacher takes 38 students, there are 6 students left without a teacher; if each teacher takes 40 students, one teacher is short of 6 students. After the labor practice, within the total rental cost limit of $2300, the school rents cars to send teachers and students back to school, with at least 1 teacher on each car. There are two types of large buses, A and B, with their passenger capacities and rental costs as shown in the table below:

|            | Bus A | Bus B |
|------------|-------|-------|
| Passenger capacity (people per bus) | 45 | 30 |
| Rental cost (yuan per bus) | 400 | 280 |

$(1)$ How many teachers and students are participating in this practice activity?  
$(2)$ When returning to school by rental car, ensuring that all teachers and students have a seat and at least 1 teacher on each car, how many cars are needed in total?  
$(3)$ How many rental car plans does the school have? What is the minimum rental cost?

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