
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 survey was conducted on the number of social practice activities of 100 students selected by simple random sampling, with the results shown in the table below:

| $n$            | $\left[0,3\right)$ | $\left[3,6\right)$ | $\left[6,9\right)$ | $\left[9,12\right)$ | $\left[12,15\right)$ | $\left[15,18\right)$ |
|----------------|-------------------|-------------------|-------------------|---------------------|----------------------|----------------------|
| Number of Boys | 7                 | 15                | 11                | 12                  | 2                    | 1                    |
| Number of Girls| 5                 | 13                | 20                | 9                   | 3                    | 2                    |

If students who have participated in social practice activities at least 12 times are called "social practice models."

$(Ⅰ)$ Considering frequency as probability, estimate how many "social practice models" are there among the 1600 students in the school?

$(Ⅱ)$ From the 8 selected "social practice models," 3 are boys and 5 are girls.

(i) Let $A$ be the event "4 selected students include both boys and girls." Find the probability of event $A$ occurring.

(ii) Let $X$ represent the number of boys among the selected "social practice models." 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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