
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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A certain interest group is studying the relationship between a local disease and the hygiene habits (classified as good or not good) of local residents. Let $A=$"suffering from the local disease" and $B=$"good hygiene habits". According to clinical statistics, $P(A|\overline{B})=\frac{3}{4}$ and $P(B|\overline{A})=\frac{12}{13}$. The probability of having good hygiene habits in the local population is $\frac{4}{5}$.  
$(1)$ Find $P\left(A\right)$ and $P\left(A|B\right)$;  
$(2)$ To further verify the judgment in $(1)$, the interest group used stratified sampling to select a sample of size $m(m\in N^{*})$ in the area. Using an independence test, they calculated $\chi ^{2}=2.640$. To increase the reliability of the test conclusion, the sample size is adjusted to $k(k\in N^{*})$ times the original size, ensuring a $99.9\%$ certainty of confirming the judgment in $(1)$. Determine the minimum value of $k$.  
Table and formula: $χ^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)},n=a+b+c+d$  
|$α$|$0.10$|$0.05$|$0.010$|$0.005$|$0.001$|  
|$x_{\alpha }$|$2.706$|$3.841$|$6.635$|$7.879$|$10.828$|

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