
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

...

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Nowadays, the country vigorously promotes the socialization and marketization of elderly care. Elderly apartments are one of the measures to meet the high-quality, diversified, and professional living and recuperation needs of the elderly. In order to provide better services to the elderly, the person in charge of a certain elderly apartment is collecting opinions from the 120 elderly residents. The types of rooms occupied by the elderly residents of this apartment are shown in the table below:

| Type of Room | Single Room | Double Room | Triple Room |
|--------------|-------------|-------------|-------------|
| Number of Residents | 36 | 60 | 24 |

$(1)$ If a stratified sampling method is used to randomly select 10 people from these 120 elderly residents based on the types of rooms they occupy, and then 4 people are randomly selected from these 10 people for inquiry, let the number of residents occupying single rooms among the randomly selected 4 people be denoted as $\xi$, find the probability distribution and mathematical expectation of $\xi$.  
$(2)$ Let double rooms and triple rooms be referred to as multiple occupancy rooms. If, during the opinion collection, it is required to group 2 residents occupying single rooms and $m$ residents (where $m > 2$ and $m \in \mathbb{N}^{*}$) occupying multiple occupancy rooms together, and the person in charge randomly selects 2 people from a group for inquiry, if the selected 2 people have the same room type, the group is labeled as $Ⅰ$, otherwise, it is labeled as $Ⅱ$. Let the probability of a certain group being labeled as $Ⅱ$ be denoted as $p$.  
$(i)$ Express $p$ using an algebraic expression involving $m$.  
$(ii)$ If a total of 5 groups are inquired, and $g(p)$ represents the probability that exactly 3 groups are labeled as $Ⅱ$, find the maximum value of $g(p)$ and the corresponding value of $m$.

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