
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 car purchase costs 144,000 yuan. Each year, the total cost for insurance and gasoline is 9,000 yuan. The repair expenses for the car are as follows: 2,000 yuan for the first year, 4,000 yuan for the second year, 6,000 yuan for the third year, ... , with the amount increasing by an arithmetic sequence each year.
(1) Let $f(n)$ represent the total cost (including the purchase cost) of using the car for $n$ years. Please establish the expression for $f(n)$.
(2) How many years of usage would make the scrapping of the car the most economical (i.e., after how many years of usage will the annual average cost be the lowest)?
(3) If the car is purchased through installments, with the first payment made one month after purchase, and equal payments made on the same day of each month, with the total amount paid off in the first year (24 months) and the monthly interest rate being 1% compounded monthly, how much should be paid to the car dealer each month (round the result to the nearest yuan, use the reference data $1.01^{24} \approx 1.27$)?

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