
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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Starting from the explosive popularity of the TV series "Tang Palace Night Banquet," a certain TV station launched the "Chinese Festival" series program, which has attracted widespread attention. A statistical platform randomly selected 100 citizens from the entire city to investigate their attitudes towards the "Chinese Festival" series program. The frequency distribution of their ages and the number of people who like the "Chinese Festival" series program are shown in the table below. (Note: The age unit is in years, and all ages are within the range of $[15, 75)$)

| Age | $[15,25)$ | $[25,35)$ | $[35,45)$ | $[45,55)$ | $[55,65)$ | $[65,75)$ |
|-----|-----------|-----------|-----------|-----------|-----------|-----------|
| Frequency | $10$ | $20$ | $30$ | $20$ | $10$ | $10$ |
| Likers | $6$ | $16$ | $26$ | $12$ | $6$ | $4$ |

$(1)$ If the age of $45$ is taken as the dividing point, complete the $2 \times 2$ contingency table below based on the above statistical data, and determine through calculation whether it can be considered that the attitude towards the "Chinese Festival" series program is related to age, with the condition that the probability of making an error does not exceed $0.01$.

| | Number of people aged 45 or older | Number of people younger than 45 | Total |
|-----|----------------------------------|-----------------------------------|-------|
| Likers | | | |
| Dislikers | | | |
| Total | | | |

$(2)$ If the age group is stratified by random sampling from the sample, selecting $8$ people from the sample with ages in $[45,75)$, and then randomly selecting $3$ people from these $8$ people, find the distribution table and mathematical expectation of the number $X$ of people aged $[55,65)$ among these $3$ people. Reference formula and data: ${χ^2}=\frac{{n{{({ad-bc})}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$, where $n=a+b+c+d$.

| $P(X^{2}\geqslant k_{0})$ | $0.05$ | $0.01$ | $0.005$ | $0.001$ |
|---------------------------|--------|--------|---------|---------|
| $k_{0}$ | $3.841$ | $6.635$ | $7.879$ | $10.828$ |

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