
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
The 25th Winter Olympic Games will be held in 2026. A certain country's sports team plans to send three athletes, A, B, and C, to participate in the freestyle skiing competition. The competition consists of a preliminary round and a final round, with two rounds in the preliminary round, and only those who win both rounds can enter the final round. It is known that the probability of athlete A winning in each round is $\frac{3}{4}$; athlete B's probabilities of winning in the first and second rounds are $\frac{4}{5}$ and $\frac{5}{8}$, respectively; athlete C's probabilities of winning in the first and second rounds are $p$ and $\frac{3}{2}-p$, respectively, where $\frac{1}{2}＜p＜\frac{3}{4}$.  
$(1)$ Determine which of A, B, and C has the highest probability of entering the final round.  
$(2)$ If the probabilities of A, B, and C all entering the final round are $\frac{5}{32}$, find the distribution and expectation of the number of athletes, denoted as $\xi$, entering the final round.

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