
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 school's debate team plans to participate in a debate on Saturday and Sunday, each led by the captain and vice-captain. It is known that the debate team has a total of 10 members (including the captain and vice-captain). Each debate requires 3 additional team members (the same team member can participate in both days of the debate, and the captain and vice-captain can only participate in one debate each). Assuming that the captain and vice-captain independently and randomly send the notification information for their respective debates to 3 out of 8 team members, and all recipients receive the information.

$(1)$ Find the probability that team member A receives the notification information for the debate sent by either the captain or the vice-captain.

$(2)$ Let the number of team members who receive the notification information for the debate sent by the captain or vice-captain be $X$. Find the probability distribution of $X$, as well as its mean and variance.

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