
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
On February 20, 2022, the Beijing Winter Olympics came to a close at the Bird's Nest, with the Chinese team achieving its best historical performance. The successful hosting of the Beijing Winter Olympics has promoted the popularization of ice and snow sports in China, leading more and more young people to fall in love with ice and snow sports. A school organized a school-wide ice and snow sports knowledge competition and compiled the scores of 100 participating students into the following frequency distribution table:

| Score Range | $[50,60]$ | $(60,70]$ | $(70,80]$ | $(80,90]$ | $(90,100]$ |
|-------------|-----------|-----------|-----------|-----------|------------|
| Frequency   | $0.1$     | $0.1$     | $0.3$     | $0.3$     | $0.2$      |

$(1)$ If the score range $(80,90]$ is defined as "good" and the score range $(90,100]$ is defined as "excellent," from the two groups of students with "good" and "excellent" scores, 5 students are randomly selected using stratified sampling. From these 5 students, 2 students are selected for a discussion. Find the probability that both students have an "excellent" score.

$(2)$ Taking the frequency of students with "excellent" scores among the 100 participating students as the probability of being selected in the school-wide knowledge competition. Now, randomly select 3 students from this school. Let $X$ denote the number of students with "excellent" scores. Find the probability distribution and the expected value of the random variable $X$.

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