
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 line with the concept of a healthy, low-carbon lifestyle, more and more people are renting bicycles for rides. The charging standard at a bicycle rental point in a tourist attraction in Hangzhou is free for each bicycle rented for no more than two hours. For rentals exceeding two hours, the charge is 2 yuan (for parts of an hour less than 1 hour, it is calculated as 1 hour, e.g., a 2.5-hour rental costs 2 yuan, a 3.5-hour rental costs 4 yuan). Individuals come to this rental point to rent bicycles for rides, each renting a bicycle once. Let the probabilities of A and B returning the bicycle within two hours be $\dfrac{1}{4}$ and $\dfrac{1}{2}$, respectively; the probabilities of returning the bicycle after two hours but within three hours be $\dfrac{1}{2}$ and $\dfrac{1}{4}$, respectively; neither of the two will rent the bicycle for more than four hours.

$(1)$ Calculate the probability that A and B pay the same rental fee;

$(2)$ Let the sum of the rental fees paid by A and B be a random variable $\eta$. Calculate the distribution and the mathematical expectation $E(\eta)$ of $\eta$.

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