
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
To understand the audience's evaluation of the skit "Pit" in the 2023 CCTV Spring Festival Gala, a certain organization randomly selected 10 audience members to rate it, as shown in the table below:

| Audience Number | 1   | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   | 10  |
|-----------------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| Rating          | 7.8 | 8.9 | 8.6 | 7.4 | 8.5 | 8.5 | 9.5 | 9.9 | 8.3 | 9.1 |

$(1)$ Find the $75$th percentile of this data set.  
$(2)$ Considering frequency as probability, now randomly select 3 audience members to evaluate the skit "Pit." Let $X$ be the number of people among the 3 selected whose rating exceeds $9.0$. Find the probability distribution, expected value, and variance of $X$.

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