
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
With the concept of a healthy and low-carbon lifestyle, more and more people are renting bicycles for leisurely rides. The charging standard of a certain bicycle rental point is that each bike is free for up to two hours each time, and the part exceeding two hours is charged $2$ dollars per hour (the part less than one hour is counted as one hour). Two individuals, A and B, independently come to rent bikes at this rental point (each renting one bike once). It is given that the probabilities of A and B returning the bike within two hours are $\frac{1}{4}$ and $\frac{1}{2}$, respectively, and the probabilities of returning the bike after two hours but within three hours are $\frac{1}{2}$ and $\frac{1}{4}$, respectively. Both A and B will not rent the bikes for more than four hours.<br/>$(1)$ Find the probability that A and B will pay the same rental fee;<br/>$(2)$ Let $\xi$ be the sum of the rental fees paid by A and B. Find the values of $P(\xi =4)$ and $P(\xi =6)$.

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