
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
The comprehensive solar detection satellite "Kua Fu I" in our country is a space science satellite developed by the second phase of the Chinese Academy of Sciences Space Science Pilot Project. The satellite has the scientific goal of "one magnetic field and two bursts", that is, to simultaneously observe the solar magnetic field and the two most intense eruptive phenomena of the sun - solar flares and coronal mass ejections. In order to understand whether a certain interest group in a school is interested in the relevant knowledge of the "Kua Fu I" detection satellite, a survey was conducted on 100 students in the interest group. It is known that among the surveyed students, boys account for 55% of the survey population, of which 45 are interested, and the rest are not interested. Among the surveyed girls, 20 are interested, and the rest are not interested.

$(1)$ Please complete the $2\times 2$ contingency table, and based on the small probability value $\alpha =0.005$ of the independence test, can it be concluded that the interest in the knowledge related to the "Kua Fu I" detection satellite is associated with the gender of the students?

\[
\begin{array}{|c|c|c|c|}
\hline
& \text{Interested} & \text{Not Interested} & \text{Total} \\
\hline
\text{Boys} & & & \\
\hline
\text{Girls} & & & \\
\hline
\text{Total} & & & \\
\hline
\end{array}
\]

$(2)$ From the 100 people in the interest group, one person is randomly selected. Let $A$ represent the event "the person selected is a boy," and $B$ represent the event "the person selected is not interested in the knowledge related to 'Kua Fu I' detection satellite." Find $P(B|A)$.

$(3)$ By randomly sampling 4 boys and 3 girls from the interested students, forming a sample of size 7, and then randomly selecting 3 people from the 7 selected, let the random variable $X$ represent the number of girls among the 3 people. Find the distribution table and mathematical expectation of $X$.

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

Critical value table:

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\alpha & 0.15 & 0.10 & 0.05 & 0.01 & 0.005 \\
\hline
x_{\alpha} & 2.072 & 2.706 & 3.841 & 6.635 & 7.879 \\
\hline
\end{array}
\]

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