
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
Rural homestays are based in the countryside, catering to the modern pursuit of getting away from the hustle and bustle, getting close to nature, and savoring the nostalgia of the countryside. In a town on the eve of the peak tourist season, in order to understand the quality of ordinary and high-quality rural homestays in each village, $8$ relatively large rural homestays were randomly selected, and the number of rooms in each homestay is shown in the table below:

| Homestay | A   | B   | C   | D   | E   | F   | G   | H   |
|----------|-----|-----|-----|-----|-----|-----|-----|-----|
| Ordinary | $16$ | $8$ | $12$ | $14$ | $13$ | $18$ | $9$ | $20$ |
| Quality  | $6$  | $16$ | $4$  | $10$ | $11$ | $10$ | $9$ | $12$ |

$(1)$ From these $8$ homestays, if $3$ homestays are randomly selected under the condition that the number of rooms in the ordinary homestays is not less than $10$ in each of the selected homestays, find the probability that the number of rooms in the quality homestays of these $3$ homestays is not less than $10$.

$(2)$ From these $8$ homestays, if $4$ homestays are randomly selected, let $X$ be the number of homestays among the selected $4$ homestays where the number of rooms in the ordinary homestays is not less than $15$. Find the probability distribution and the mathematical expectation of $X$.

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