
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 city uses a lottery system for assigning car permits, with 300,000 people participating in the lottery and 30,000 permits available each month.
1. If those who win the lottery each month exit the lottery, and those who do not win continue in the following month's lottery, with an additional 30,000 new participants added each month, how long on average does it take for each person to win a permit?
2. Under the conditions of part (1), if the lottery authority can control the proportion of winners such that in the first month of each quarter the probability of winning is $\frac{1}{11}$, in the second month $\frac{1}{10}$, and in the third month $\frac{1}{9}$, how long on average does it take for each person to win a permit?

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