
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

...

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A certain school's "Soccer Club" conducted a survey to investigate whether students' preference for soccer is related to their gender. They randomly selected $80k(k\in N^{*})$ students from the entire school. If the ratio of boys to girls among those selected is $5:3$, and the proportion of boys who like soccer is $\frac{3}{5}$, and the proportion of girls who like soccer is $\frac{1}{3}$. After calculations, it was found that there is a $95\%$ confidence that liking soccer is related to gender, but there is not a $99\%$ confidence that liking soccer is related to gender. 

$(1)$ Please complete the contingency table below and find the value of $k$:

|         | Likes Soccer | Does Not Like Soccer | Total |
|---------|--------------|----------------------|-------|
| Boys    |              |                      |       |
| Girls   |              |                      |       |
| Total   |              |                      |       |

$(2)$ Considering frequencies as probabilities and using the sample to estimate the population, if 4 boys are randomly selected from the entire male student population, and $X$ represents the number of boys who like soccer, find the distribution table and the expected value of $X$.

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

| $P(\chi^{2}\geq k_{0})$ | $0.10$ | $0.05$ | $0.01$ | $0.001$ |
|-------------------------|--------|--------|--------|---------|
| $k_{0}$                 | $2.706$| $3.841$| $6.635$| $10.828$|

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