
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 newly established car rental company purchased a batch of new cars for 1,020,000 yuan at the beginning of this year. During the usage period, they have an annual income of 200,000 yuan, which is immediately invested in operations. The maintenance and repair costs for the first year are 10,000 yuan, and from the second year onwards, these costs increase by 10,000 yuan each year. After usage, this batch of cars can be sold at a price of $(30 - \frac{1}{2}x)$ million yuan at the end of the x-th year (x is a natural number and x ≤ 20).

(I) Find the analytical expression of the total profit function y (in million yuan) of the company at the end of the x-th year, and find its maximum value;

(II) To maximize economic efficiency, i.e., to maximize the average annual profit, when should the company sell this batch of cars? Explain your reasoning.

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