
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 May of the second semester of the 11th grade, a school held a volleyball competition for the entire grade, with a total of 21 teams, including 20 student teams and one teacher team. The competition rules were as follows: The 20 student teams competed in two rounds of elimination matches, selecting 5 student teams to directly enter the quarterfinals. Then, from the 15 student teams that were eliminated, 2 student teams were randomly selected through a drawing. These 7 student teams, along with the teacher team, participated in the quarterfinal elimination matches, and the final champion was determined after three rounds of elimination matches. If the probability of winning between student teams is $\frac{1}{2}$ for both sides, and the probability of the teacher team winning against student teams is $\frac{3}{5}$.
$(1)$ Find the probability that Class A advances directly to the quarterfinals in the first two rounds of elimination matches (without drawing).
$(2)$ Let $X$ represent the number of rounds the teacher team participates in the competition. Find the probability distribution and expectation of $X$.

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