
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]

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[ACTION NAME]

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# Your action step 2

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...

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The popularity of Zibo barbecue coincides with the public's emotional impulse of "leaving when you say you want to leave." Food is not only a part of life but also a reflection of social emotions. With the increasing frequency of population movement between cities, giving oneself a spontaneous trip is the choice of many young people nowadays. In order to understand the attitudes of young people towards Zibo barbecue, a random survey of 200 young people was conducted, and the incomplete $2\times 2$ contingency table below shows the statistical data obtained (unit: people):

|         | Very Like | Average Feeling | Total |
|---------|-----------|-----------------|-------|
| Male    |           | $a$             |       |
| Female  | $2a$      |                 | $100$ |
| Total   |           | $70$            |       |

$(1)$ Find the value of $a$ and determine whether there is a $95\%$ confidence that the attitudes of young people towards Zibo barbecue are related to gender.  
$(2)$ Select $4$ males and $3$ females from the sample as representatives, among whom $2$ males and $2$ females very much like Zibo barbecue. Now, select $3$ males and $2$ females from these $7$ representatives for further communication. Let $\xi$ be the number of people among these $5$ who very much like Zibo barbecue. Find the distribution table and the mathematical expectation $E(\xi)$ of $\xi$.  
Reference formula: ${\chi}^{2}=\frac{n{(ad-bc)}^{2}}{(a+b)(c+d)(a+c)(b+d)}$, where $n=a+b+c+d$.  
Reference data:

| $P(\chi^{2}\geq k_{0})$ | $0.1$ | $0.05$ | $0.01$ |
|-------------------------|-------|--------|--------|
| $k_{0}$                 | $2.706$ | $3.841$ | $6.635$ |

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