
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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The Education Bureau of a district is preparing to organize a safety knowledge competition. A school plans to select $200$ students for the safety knowledge test using a stratified sampling method based on gender. Let $A=$ "male," $B=$ "score exceeds $85$," and $P({A|\overline{B}})=\frac{2}{5}$, $P({B|\overline{A}})=\frac{5}{8}$, $P(B)=\frac{3}{4}$. <br/>$(1)$ Complete the $2\times 2$ contingency table below and based on the independence test with a small probability value of $\alpha =0.001$, can it be inferred that the level of safety knowledge of students in the school is related to gender?<br/><table><tbody><tr><td rowspan="2">Gender</td><td colspan="2">Level of Safety Knowledge</td><td rowspan="2">Total</td></tr><tr><td>Number of people with a score not exceeding $85$</td><td>Number of people with a score exceeding $85$</td></tr><tr><td>Male</td><td></td><td></td><td></td></tr><tr><td>Female</td><td></td><td></td><td></td></tr><tr><td>Total</td><td></td><td></td><td></td></tr></tbody></table>$(2)$ The school plans to select the top two male and female students respectively to represent the school in the district-level competition. It is known that the probability of a male student winning is $\frac{3}{4}$, and the probability of a female student winning is $\frac{2}{3}$. Let $X$ be the number of students from the school who win, find the distribution table and mathematical expectation of $X$.<br/>The table below shows some common small probability values and corresponding critical values for the $\chi ^{2}$ independence test.<br/><table><tbody><tr><td width="74">$\alpha $</td><td width="93" align="center">$0.1$</td><td width="93" align="center">$0.05$</td><td width="85" align="center">$0.01$</td><td width="84" align="center">$0.005$</td><td width="82" align="center">$0.001$</td></tr><tr><td>$x_{a}$</td><td align="center">$2.706$</td><td align="center">$3.841$</td><td align="center">$6.635$</td><td align="center">$7.879$</td><td align="center">$10.828$</td></tr></tbody></table>

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