
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
There is a road with 10 bus stops, \( A_{0}, A_{1}, A_{2}, \cdots, A_{9} \). The distance between any two adjacent stops is \( a \) kilometers. A vehicle starts from \( A_{0} \), stops at each bus stop exactly once, and finally returns to the starting point \( A_{0} \). The vehicle does not necessarily stop at the stations in the sequential order \( A_{1}, A_{2}, A_{3}, \cdots \) (e.g., it can go from \( A_{3} \rightarrow A_{5} \rightarrow A_{7} \rightarrow \cdots \)). What is the maximum possible distance the vehicle can travel? What is the minimum possible distance the vehicle can travel? Provide reasoning.

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