
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
A certain TV station randomly selected $100$ viewers to evaluate a TV program in order to understand the evaluation of the same TV program by viewers of different genders. It is known that the ratio of the number of "male" to "female" viewers selected is $9:11$. The evaluation results are divided into "like" and "dislike", and some evaluation results are organized in the table below.

| Gender | Like | Dislike | Total |
|--------|------|---------|-------|
| Male   | $15$ |         |       |
| Female |      |         |       |
| Total  | $50$ |         | $100$  |

$(1)$ Based on the given data, complete the $2\times 2$ contingency table above. According to the independence test with $\alpha = 0.005$, can it be concluded that gender is related to the evaluation results?  
$(2)$ The TV station plans to expand the male audience market. Now, using a proportional stratified sampling method, $3$ viewers are selected from the male participants for a program "suggestions" solicitation reward activity. The probability that a viewer who evaluated "dislike" has their "suggestion" adopted is $\frac{1}{4}$, and the probability that a viewer who evaluated "like" has their "suggestion" adopted is $\frac{3}{4}$. The reward for an adopted "suggestion" is $100$ dollars, and for a non-adopted "suggestion" is $50$ dollars. Let $X$ be the total prize money obtained by the $3$ viewers. Find the distribution table and the expected value of $X$.  
Given: ${\chi}^{2}=\frac{n{(ad-bc)}^{2}}{(a+b)(c+d)(a+c)(b+d)}$  
| $\alpha$ | $0.010$ | $0.005$ | $0.001$ |
|----------|---------|---------|---------|
| $x_{\alpha}$ | $6.635$ | $7.879$ | $10.828$ |

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