
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
Calculate the total cost of owning a certain type of car for \( n \) years, given that the initial purchase cost is 100,000 Yuan, the annual cost for insurance and fuel is 9,000 Yuan, and the maintenance costs for the car increase by 2,000 Yuan each year, starting at 2,000 Yuan during the first year.
1. Derive the expression for the total cost \( S_n \) after \( n \) years.
2. Determine after how many years it would be most economical to consider the car a write-off (i.e., the lowest average annual cost).

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