
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
Internet users evaluate the quality of a certain special agricultural product sales service on an e-commerce platform. Each participating online shopper selects one of "good review, neutral review, bad review" for evaluation. A sample of $20,000$ people was randomly selected from the participants, and the statistics were conducted on two age groups: "below $50$ years old" and "$50$ years old and above (including $50$ years old)", obtaining the number of people giving "good review, neutral review, bad review" as shown in the table below.

| Age Group           | Good Review | Neutral Review | Bad Review |
|---------------------|-------------|----------------|------------|
| Below 50 years old  | 10,000      | 2,000          | 2,000      |
| 50 years old and above (including 50 years old) | 2,000       | 3,000          | 1,000      |

$(1)$ Based on this sample of $20,000 people to estimate the population, one person is randomly selected from the participating evaluators each time. If "good review" is selected, the selection stops; otherwise, it continues until a "good review" is selected, but the maximum number of selections does not exceed $5$ times. Find the probability of selecting $5$ times;  
$(2)$ From the evaluators who gave a "neutral review", $10$ people are selected using stratified sampling. Then, $3$ people are randomly selected from these $10$ people. Let $X$ be the number of people under $50$ years old among the $3$ selected people. Find the probability distribution and the mathematical expectation $E(X)$ of $X$.

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