
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
A certain city's health department randomly selected $16$ universities in the city to conduct quantitative evaluations on the "qualification of procurement channels" and "food safety" of their cafeterias. The cafeteria scores of all universities are within the range of $7$ to $10$ points. The table below records their scores:

| Score | $[7,8)$ | $[8,9)$ | $(9,10)$ |
|-------|---------|---------|---------|
| Number of Universities | $4$ | $8$ | $4$ |

$(1)$ Now, $3$ universities are randomly selected from the $16$ universities. Find the probability that at most $1$ university has a cafeteria score of at least $9$ points.  
$(2)$ Using the cafeteria score data of these $16$ universities, estimate the cafeteria scores of all universities in the city. If $3$ universities are randomly selected from all the universities in the city, let $X$ be the number of universities with cafeteria scores of at least $9$ points. Find the distribution of $X$.

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