
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
Following the concept of a healthy, low-carbon lifestyle, an increasing number of people are renting bicycles for cycling tours. A particular bicycle rental point charges no fee for rentals that do not exceed two hours, and for rentals that exceed two hours, the charging standard is 2 yuan per hour (with fractions of an hour calculated as a full hour). Suppose two individuals, A and B, each rent a bicycle once. The probability that A and B return their bicycles within two hours is $\frac{1}{4}$ and $\frac{1}{2}$, respectively. The probability that they return their bicycles between two and three hours is $\frac{1}{2}$ and $\frac{1}{4}$, respectively. Neither A nor B will rent a bicycle for more than four hours.

(I) Calculate the probability that the bicycle rental fees paid by A and B are the same.
(II) Let $\xi$ be the random variable representing the sum of the bicycle rental fees paid by A and B. Find the distribution of $\xi$ and its mathematical expectation $E_{\xi}$.

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