
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
From five people, A, B, C, etc., three people are randomly selected to do passing training. The training rule is to determine one person to pass the ball for the first time. Each time the ball is passed, the passer will pass the ball to either of the other two people with equal probability, and the ball must be passed each time.  
$(1)$ Let the random variable $X$ represent the number of people selected from A, B, and C. Find the distribution of $X$.  
$(2)$ If exactly A, B, and C are selected to do passing training, and A passes the ball for the first time, let the probability that the ball is in A's hands after $n$ passes be $p_{n}$, where $n=1, 2, 3, \cdots$.  
① Write down the values of $p_{1}$, $p_{2}$, and $p_{3}$.  
② Find the relationship between $p_{n+1}$ and $p_{n}$ for $n \in \mathbb{N}^{*}$, and find $p_{n}$ for $n \in \mathbb{N}^{*}$.

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