
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
An enterprise is implementing new measures and formulating new plans to demonstrate greater achievements in promoting the new journey of Chinese-style modernization, focusing on enhancing employee dedication and management level. An evaluation of employee dedication and management level is conducted by selecting 200 people from the enterprise. Among them, 50 people are satisfied with both employee dedication and management level. The number of people satisfied with employee dedication accounts for 40% of the total, and the number of people satisfied with management level accounts for 45% of the total. 

$(1)$ Complete a $2\times 2$ contingency table for the evaluation of employee dedication and management level. Based on the independence test with a small probability value of $\alpha = 0.01$, can it be concluded that satisfaction with employee dedication is related to satisfaction with management level?

$(2)$ If frequencies are considered as probabilities, randomly select 3 people from the enterprise to participate in this evaluation. Let the random variable $X$ represent the number of people satisfied with both employee dedication and management level. Find the distribution table and mathematical expectation of $X$.

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

| $\alpha$ | $0.050$ | $0.010$ | $0.001$ |
|----------|---------|---------|---------|
| $x_{\alpha}$ | $3.8410$ | $6.635$ | $10.828$ |

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