
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
In Hangzhou scenic area, there is a bicycle rental company with three stations: A, B, and C. Customers can rent bicycles from any station and return them to any station. According to statistics, the pattern of renting and returning bicycles is as follows:
1) 30% of customers who rent bicycles at station A return them at A, 20% at B, and 50% at C;
2) 70% of customers who rent bicycles at station B return them at A, 10% at B, and 20% at C;
3) 40% of customers who rent bicycles at station C return them at A, 50% at B, and 10% at C.
Let $P(XY)$ denote the probability of a bicycle being rented at station X and returned at station Y, and $P(XY)P(YZ)$ denote the probability of a bicycle being rented at station X, returned at station Y, and then rented again at Y and returned at Z. According to the rules specified above,
(1) Find $P(CC)$;
(2) Find $P(AC)P(CB)$;
(3) Suppose a bicycle is rented from station A, find the probability $P$ that this bicycle is returned to a station, rented out again, and then returned to station A.

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