
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
In the autumn of 2023, mycoplasma pneumonia was prevalent in various parts of China, with the main infected groups being teenagers and the elderly. The infectious disease department of a certain city hospital randomly surveyed $200$ elderly people aged $70$ and above who had sought medical treatment in various hospitals in the city during a certain period, and organized the survey results as shown in the table below:

|                      | With Chronic Disease | Without Chronic Disease | Total |
|----------------------|----------------------|-------------------------|-------|
| Not infected         | $60$                 | $80$                    | $140$ |
| with mycoplasma pneumonia | $40$                 | $20$                    | $60$  |
| Total                | $100$                | $100$                   | $200$ |

$(1)$ Can we be $99.5\%$ confident that mycoplasma pneumonia infection in elderly people aged $70$ and above is related to having chronic diseases?

$(2)$ Now, $6$ elderly people with mycoplasma pneumonia are sampled using stratified sampling method from the $60$ elderly people. Then $4$ of the $6$ sampled individuals have chronic diseases, and $2$ do not. Let the $4$ individuals with chronic diseases be denoted as $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, and the $2$ individuals without chronic diseases be denoted as $B_{1}$, $B_{2}$. Now, $2$ individuals are randomly selected from the $6$ sampled individuals in $15$ possible ways. The specific cases are as follows:

$A_{1}A_{2}$, $A_{1}A_{3}$, $A_{1}A_{4}$, $A_{1}B_{1}$, $A_{1}B_{2}$, $A_{2}A_{3}$, $A_{2}A_{4}$, $A_{2}B_{1}$, $A_{2}B_{2}$, $A_{3}A_{4}$, $A_{3}B_{1}$, $A_{3}B_{2}$, $A_{4}B_{1}$, $A_{4}B_{2}$, $B_{1}B_{2}$.

Among these $15$ cases, there are $8$ cases where exactly $1$ person has a chronic disease, such as $A_{1}B_{1}$, $A_{1}B_{2}$, $A_{2}B_{1}$, $A_{2}B_{2}$, $A_{3}B_{1}$, $A_{3}B_{2}$, $A_{4}B_{1}$, $A_{4}B_{2}$. Therefore, the probability of selecting $2$ individuals with exactly $1$ person having a chronic disease is $P=\frac{8}{15}$.

Given table:

| $P(K^{2}\geqslant k)$ | $0.15$ | $0.10$ | $0.05$ | $0.025$ | $0.010$ | $0.005$ | $0.001$ |
|-----------------------|--------|--------|--------|---------|---------|---------|---------|
| $k$                   | $2.072$ | $2.706$ | $3.841$ | $5.024$ | $6.635$ | $7.879$ | $10.828$ |

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

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