
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 middle school conducted a survey on students who study science and engineering courses. Students who study science and engineering courses independently for more than $6$ hours per week are called "science enthusiasts," otherwise they are called "non-science enthusiasts." A random sample of $100$ people was selected for analysis from the survey results, and the data is shown in the table below:

|         | Science Enthusiasts | Non-Science Enthusiasts | Total |
|---------|---------------------|-------------------------|-------|
| Male    | $24$                | $36$                    | $60$  |
| Female  | $12$                | $28$                    | $40$  |
| Total   | $36$                | $64$                    | $100$ |

$(1)$ Based on the independence test with $\alpha = 0.010$, can it be concluded that "science enthusiasts" are related to gender?  
$(2)$ In artificial intelligence, $L(B|A) = \frac{P(B|A)}{P(\overline{B}|A)}$ is commonly used to represent the odds of event $B$ occurring given that event $A$ has occurred, known as the likelihood ratio in statistics. Now, randomly select a student from the school, where $A$ represents "the selected student is a non-science enthusiast" and $B$ represents "the selected student is a male." Using the sample data, estimate the value of $L(B|A)$.  
$(3)$ Now, from the sample of "science enthusiasts," select a group of $6$ people using stratified sampling. From the selected $6$ people, randomly choose $3$ people to participate in a science knowledge competition. Find the probability distribution table of the number of male students $X$ among these $3$ people and calculate the mathematical expectation.

Reference data and formulas:

| $\alpha$ | $0.050$ | $0.010$ | $0.001$ |
|----------|---------|---------|---------|
| $x_{\alpha}$ | $3.841$ | $6.635$ | $10.828$ |

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

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