
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
According to the actual development of a certain virus mutation, significant adjustments have been made to the prevention and control measures in a certain area. Among them, whether the elderly are vaccinated has attracted much attention. In order to understand whether the elderly in a certain area have been vaccinated, a simple random sampling method was used to survey 500 elderly people in the area, and the results are as shown in the table below:

| Gender/Vaccination Status | Male | Female |
|---------------------------|------|--------|
| Not Vaccinated            | 20   | 10     |
| Vaccinated                | 230  | 240    |

$(1)$ Estimate the proportion of vaccinated elderly people in the area;  
$(2)$ Can we be $99\%$ confident that whether the elderly in the area have been vaccinated is related to gender?  
Attachment: (Reference formula: ${K^2}=\frac{{n{{(ad-bc)}^2}}}{{(a+b)(c+d)(a+c)(b+d)}}$, where $n=a+b+c+d$)  

| $P(K^{2}\geqslant k_{0})$ | $0.100$ | $0.050$ | $0.010$ | $0.005$ |
|---------------------------|---------|---------|---------|---------|
| $k_{0}$                   | $2.706$ | $3.841$ | $6.635$ | $7.879$ |

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