
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
In a learning platform, the scoring rules for the "challenge quiz" are as follows: Each player can participate in one round of the "challenge quiz" activity per day. In each round, the player needs to answer several questions in sequence. When the player accumulates $3$ correct answers, the quiz stops, and the player receives $10$ points; or when the player accumulates $2$ wrong answers, the quiz stops, and the player receives $8$ points. Assuming that player A has a probability of $p$ of answering each question correctly, where $0 < p < 1$.

$(1)$ Let $X$ denote the number of questions player A answers in one round of the "challenge quiz." Find the distribution of $X$.

$(2)$ If $p = \frac{2}{3}$, let $Y$ be the "total points player A earns from participating in the 'challenge quiz' activity for $9$ consecutive days." Find $E(Y)$.

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