
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
Three table tennis associations, A, B, and C, dispatch 3, 1, and 2 players respectively to participate in a competition. The players from association A are numbered $A\_1$, $A\_2$, $A\_3$, the player from association B is numbered $A\_4$, and the players from association C are numbered $A\_5$ and $A\_6$. If two players are randomly selected from these six to participate in a doubles match:

1. List all possible outcomes using the given numbers.
2. Calculate the probability that at least one player from association C participates in the doubles match.
3. Calculate the probability that the two players selected for the doubles match come from the same association.

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