
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]


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In 2022, the Winter Olympics, which attracted worldwide attention, was held in Beijing. The Winter Olympics mascots "Bing Dung Dung" and "Xue Rong Rong" have cute appearances and rich symbolism. Since their debut, they have received continuous praise and are deeply loved by people from various countries. A media outlet in a certain city conducted a survey among elementary school students in the city to determine whether they like these two mascots. The contingency table is as follows (unit: people):

| Gender | Like | Dislike | Total |
|--------|------|---------|-------|
| Boys   | 30   | 20      | 50    |
| Girls  | 40   | 10      | 50    |
| Total  | 70   | 30      | 100   |

$(1)$ According to the independent test of small probability value $\alpha = 0.01$, can it be inferred whether liking the mascots is related to gender?

$(2)$ Now, using the proportional allocation method, $5$ people are randomly sampled from the male students in the sample, and then $3$ people are randomly selected from these $5$ people. Let $X$ be the number of people who like the mascots among the $3$ selected people. Find the distribution table and mean of $X$.

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

| $\alpha$ | $0.1$ | $0.05$ | $0.01$ |
|----------|-------|--------|--------|
| $x_{\alpha}$ | $2.706$ | $3.841$ | $6.635$ |

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