
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 agricultural development enterprise plans to carry out the activity of "claiming a plot of land and inviting you to be a farm owner." The enterprise leases the farm in the form of a micro pastoral garden to the public for claiming. The claimed land is managed by the enterprise's professional staff, and the claimants can visit at any time to experience farming culture, with all harvests belonging to the claimants. A consulting company conducted a survey on the willingness to participate in the activity, randomly selecting 100 valid questionnaires. Some of the statistical data are shown in the table below:

| Gender | Willing to Participate | Not Willing to Participate | Total |
|--------|------------------------|---------------------------|-------|
| Male   | 48                     |                          | 60    |
| Female |                        | 18                       |       |
| Total  |                        |                           | 100   |

$(1)$ Please complete the above $2\times 2$ contingency table. Based on the small probability value $\alpha = 0.01$, conduct an independence test to analyze whether males are more willing to participate in the activity than females.  
$(2)$ To gain a more detailed understanding of the situation, a group of people who are not willing to participate in the activity were selected from the 100 valid questionnaires to form an observation group. The observation group consists of 4 males and 3 females. Three individuals are selected from the observation group as free trial participants. Let the number of males among the free trial participants be $X$. Find the probability distribution and mathematical expectation of $X$.  
Given: ${\chi^2}=\frac{{n{{(ad-bc)}^2}}}{{(a+b)(c+d)(a+c)(b+d)}}$, $n=a+b+c+d$.  
The table below provides some common small probability values and corresponding critical values for the $\chi^2$ independence test.

| $\alpha$ | $0.1$ | $0.05$ | $0.01$ | $0.005$ | $0.001$ |
|----------|-------|--------|--------|---------|---------|
| $x_{\alpha}$ | $2.706$ | $3.841$ | $6.635$ | $7.879$ | $10.828$ |

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