
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
To investigate whether the cervical spine disease of employees in an industrial park is related to the nature of their work, a random sample survey was conducted on 50 employees in the industrial park to determine if they have cervical spine disease. The contingency table obtained is as follows:

|          | Has Cervical Spine Disease | Does Not Have Cervical Spine Disease | Total |
|----------|----------------------------|--------------------------------------|-------|
| White-collar |          | $5$ |          |
| Blue-collar  | $10$     |                                      |          |
| Total    |          |                                      | $50$   |

It is known that the probability of randomly selecting a person with cervical spine disease from all 50 people is $\frac{3}{5}$.

$(1)$ Please complete the table above.
$(2)$ Based on the independence test with a small probability value of $\alpha =0.005$, analyze whether cervical spine disease is related to the nature of work.

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

| $\alpha$ | $0.15$ | $0.10$ | $0.05$ | $0.025$ | $0.010$ | $0.005$ | $0.001$ |
|----------|--------|--------|--------|---------|---------|---------|---------|
| $x_{\alpha}$ | $2.072$ | $2.706$ | $3.841$ | $5.024$ | $6.635$ | $7.879$ | $10.828$ |

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