
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 certain parking lot charges temporary parking fees based on the duration of parking. The charging standard is that each car is free for up to half an hour of parking, and any part exceeding half an hour is charged $3$ dollars per hour (rounded up to the nearest hour). Now, two individuals, A and B, are parking temporarily in the parking lot. Their parking durations do not affect each other and are both within $2.5$ hours.  
$(1)$ If the probabilities of A's parking duration falling within three time intervals - not exceeding half an hour, exceeding half an hour but not exceeding $1.5$ hours, and exceeding $1.5$ hours but not exceeding $2.5$ hours - are the same, and the probabilities of B's parking duration falling within these three intervals are also the same, find the probability that the total parking fees paid by A and B amount to $6$ dollars.  
$(2)$ If the probabilities of A and B parking for more than half an hour but not exceeding $1.5$ hours are $\frac{1}{4}$ and $\frac{1}{3}$ respectively, and the probabilities of A and B parking for more than $1.5$ hours but not exceeding $2.5$ hours are $\frac{5}{12}$ and $\frac{1}{6}$ respectively, find the probability that the parking fees paid by A and B are different.

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