
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 middle school organized all students to go to a labor practice base to carry out labor practice activities. In this activity, if each teacher leads 30 students, there will be 7 students left without a teacher; if each teacher leads 31 students, one teacher will be short of 1 student. There are two types of buses, A and B, with their passenger capacities and rental fees as shown in the table below. The school plans that the total rental fee for this labor practice activity does not exceed 3000 yuan.

|          | Type A Bus | Type B Bus |
|----------|------------|------------|
| Capacity | 35         | 30         |
| Rental   | 400        | 320        |

$(1)$ How many teachers and students are participating in this labor practice activity?  
$(2)$ If each teacher is responsible for one bus, what are the possible rental plans for the buses?  
$(3)$ Under the conditions of $(2)$, which rental plan should be chosen to save money?

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