
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
Let the number of athletes in the three table tennis associations A, B, and C be $27$, $9$, and $18$ respectively. Initially, $6$ athletes are selected using a stratified sampling method from these three associations to form a team for a competition.<br/>$(Ⅰ)$ Find the number of athletes to be selected from each of the three associations;<br/>$(Ⅱ)$ The $6$ selected athletes are labeled as $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, $A_{5}$, $A_{6}$. Two athletes are randomly selected from these $6$ athletes to participate in a doubles match.<br/>$(i)$ List all possible outcomes with the given labels;<br/>$(ii)$ Let $A$ be the event "at least one of the two athletes labeled $A_{5}$ and $A_{6}$ is selected". Find the probability of event $A$ occurring.

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