
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 certain company's products are divided into first-class and second-class based on quality. The company plans to renovate its existing production equipment. In order to analyze the effects before and after the equipment renovation, a large number of products produced before and after the renovation were randomly sampled, with 200 products selected as samples. The quality statistics of the products are shown in the table below:

|              | First-class | Second-class | Total |
|--------------|-------------|--------------|-------|
| Before renovation | 120         | 80           | 200   |
| After renovation  | 150         | 50           | 200   |
| Total        | 270         | 130          | 400   |

$(1)$ Determine whether it can be considered that the quality of this type of product produced by the company is related to the equipment renovation, under the condition that the probability of making an error does not exceed $0.01$;<br/>$(2)$ Using the stratified sampling method, 5 products were obtained from the products before the equipment renovation, including 3 first-class and 2 second-class products. Now, 3 products are randomly selected from these 5 products. Let $X$ be the number of selected first-class products. Find the distribution table of $X$ and the mean $E(X)$.<br/>Given: ${K}^{2}=\frac{n{(ad-bc)}^{2}}{(a+b)(c+d)(a+c)(b+d)}$.<br/>The table below shows the values of $P(K^{2}\geqslant k_{0})$ for different significance levels:

| $P(K^{2}\geqslant k_{0})$ | $0.050$ | $0.010$ | $0.001$ |
|---------------------------|---------|---------|---------|
| $k_{0}$                   | $3.841$ | $6.635$ | $10.828$|

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