
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
It is often said that "disease enters through the mouth," but in fact, hands are the real culprit, serving as the transportation tool between germs and the mouth. Washing hands is one of the simplest and most effective measures to prevent infectious diseases. Maintaining clean and hygienic hands can effectively reduce the risk of infection with the novel coronavirus. Proper handwashing should follow the "seven-step handwashing method," which can be summarized as "inner, outer, interlock, thumb, large, stand, and wrist," with each word representing a step. Before the school resumes, a certain school randomly selected 100 students for an online test to assess their mastery of the "seven-step handwashing method," with a maximum score of 10 points. The frequency distribution table of specific scores is as follows:

| Score   | 4  | 5  | 6  | 7  | 8  | 9  | 10 |
|---------|----|----|----|----|----|----|----|
| Female  | 2  | 9  | 14 | 13 | 11 | 5  | 4  |
| Male    | 3  | 5  | 7  | 11 | 10 | 4  | 2  |

$(1)$ Using 7 points as the boundary, divide the students' mastery of the "seven-step handwashing method" into two categories: students who scored below 7 points are considered "not mastered," while those who scored 7 points or higher are considered "basically mastered." Complete the $2\times 2$ contingency table below and determine whether there is a 95% confidence that the students' mastery of the "seven-step handwashing method" is related to gender.

|         | Not Mastered | Basically Mastered | Total |
|---------|--------------|--------------------|-------|
| Female  |              |                    |       |
| Male    |              |                    |       |
| Total   |              |                    |       |

$(2)$ From the students who participated in the online test and scored at least 9 points, 9 were female and 6 were male. Using stratified sampling by gender, 10 students were selected, with 6 females and 4 males. From the 10 students, 3 were randomly selected. Let $X$ be the number of females selected. Find the distribution table and expectation of $X$.

Given: ${K^2}=\frac{{n{{({ad-bc})}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}(n=a+b+c+d)$. Critical value table:

| $P(K^{2}\geqslant k)$ | 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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