
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 car company randomly surveyed the owners of $20n(n\in N^{*})$ new energy vehicles produced by the company in a certain month this year. The results were tabulated in a $2\times 2$ contingency table. After calculation, $\chi ^{2}\approx 5.556$ was obtained.

|         | Like   | Dislike | Total  |
|---------|--------|---------|--------|
| Male    | $10n$  |         | $12n$  |
| Female  |        | $3n$    |        |
| Total   | $15n$  |         |        |

$(1)$ Complete the table and find the value of $n$. Determine with what confidence it can be said that the preference of car consumers for new energy vehicles is related to gender.

$(2)$ Using proportional allocation stratified sampling, 12 people are randomly selected from those who like and dislike new energy vehicles. From the selected 12 people, 4 people are further selected. Let $X$ be the number of people among the selected 4 who dislike new energy vehicles. Find the probability distribution and mathematical expectation of $X$.

Given: ${\chi}^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$, where $n=a+b+c+d$.

| $P(x^{2}\geqslant k)$ | $0.15$ | $0.10$ | $0.05$ | $0.025$ | $0.010$ | $0.005$ | $0.001$ |
|-----------------------|--------|--------|--------|---------|---------|---------|---------|
| $k$                   | $2.072$| $2.706$| $3.841$| $5.024$ | $6.635$ | $7.879$ | $10.828$|

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