
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 plans to organize a collective study tour for teachers and students by renting a total of $6$ buses at a cost not exceeding $2300$ yuan. If each teacher leads a group of $12$ students, there will be $8$ students left without a teacher. If each teacher leads a group of $13$ students, one teacher will be short of $6$ students. There are two types of large buses, A and B, with their passenger capacity and rental fees as shown in the table below:

|            | Type A Bus | Type B Bus |
|------------|------------|------------|
| Passenger Capacity (people/bus) | 40 | 30 |
| Rental Fee (yuan/bus) | 400 | 320 |

$(1)$ How many teachers and students are participating in this study tour?  
$(2)$ How many rental options does the school have? What is the minimum rental cost?

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