
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 survey was conducted on the weekly exercise time of 900 students in a certain school, including 540 boys and 360 girls. Using stratified sampling based on gender, 60 students were selected from the 900 students for analysis. The statistical data is as follows (exercise time unit: hours).

Statistics of exercise time for boys:

| Exercise Time (hours) | $[0,3)$ | $[3,6)$ | $[6,9)$ | $[9,12)$ | $[12,15)$ |
|-----------------------|---------|---------|---------|----------|-----------|
| Number of Students    | $x$     | $9$     | $8$     | $12$     | $4$       |

Statistics of exercise time for girls:

| Exercise Time (hours) | $[0,3)$ | $[3,6)$ | $[6,9)$ | $[9,12)$ | $[12,15)$ |
|-----------------------|---------|---------|---------|----------|-----------|
| Number of Students    | $y$     | $10$    | $5$     | $2$      | $1$       |

$(1)$ Calculate the values of $x$ and $y$. If students who exercise for at least 6 hours per week are called "exercise enthusiasts" and students who exercise less than 6 hours per week are called "non-exercise enthusiasts," fill in the $2\times 2$ contingency table below based on the above statistical data. Can it be considered that "exercise enthusiasts" are related to gender, given that the probability of making an error does not exceed $0.025$?

|              | Boys | Girls | Total |
|--------------|------|-------|-------|
| Enthusiasts  |      |       |       |
| Non-enthusiasts |      |       |       |
| Total        |      |       |       |

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

| $P(K^{2}\geqslant k_{0})$ | $0.10$ | $0.05$ | $0.025$ | $0.010$ | $0.005$ |
|---------------------------|--------|--------|---------|---------|---------|
| $k_{0}$                   | $2.706$| $3.841$| $5.024$ | $6.635$ | $7.879$|

$(2)$ From the sample of 60 students, 3 students are randomly selected from those who exercise in the interval $[0,3)$. Let the random variable $\xi$ denote the number of boys selected. Find the distribution table and the expected value of $\xi$.

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