
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 disease can be divided into two types, $A$ and $B$. In order to determine whether the type of the disease is related to the gender of the patients, a survey was conducted on 1800 patients with the disease in a certain region. It was found that the number of female patients is half of the number of male patients, the number of male patients with type $A$ disease is two-thirds of the number of male patients, and the number of female patients with type $A$ disease is three-fourths of the number of female patients.

$(Ⅰ)$ Complete the following $2\times 2$ contingency table based on the given information:

| Gender | Type $A$ | Type $B$ | Total |
|--------|----------|----------|-------|
| Male   |          |          |       |
| Female |          |          |       |
| Total  |          |          |       |

$(Ⅱ)$ Based on the completed $2\times 2$ contingency table in $(Ⅰ)$, analyze whether the type of disease is related to gender using a $\chi^2$ independence test with a significance level of $\alpha = 0.001$.

$(Ⅲ)$ A team conducted a trial to develop a vaccine for type $A$ disease. During the trial, at most 2 vaccination cycles were arranged, with each person receiving 3 doses per cycle at a cost of 9 yuan per dose. The probability that the vaccine developed by the team produces antibodies after each dose is $\frac{2}{3}$. If at least 2 occurrences of antibodies are observed within the first cycle, the trial ends after that cycle; otherwise, it proceeds to the second cycle. Let $\xi$ denote the cost for one person to receive the vaccine in the trial. Find $E(\xi)$.

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

| $\alpha$ | $0.100$ | $0.050$ | $0.010$ | $0.005$ | $0.001$ |
|----------|---------|---------|---------|---------|---------|
| $x_{\alpha}$ | $2.706$ | $3.841$ | $6.635$ | $7.879$ | $10.828$ |

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