
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 factory produces products through three processes, which do not affect each other. It is known that the defective rates of the three processes are $\frac{1}{10}$, $\frac{1}{11}$, $\frac{1}{12}$, respectively.
$(1)$ Find the defective rate of the product.
$(2)$ From a large number of products produced by the factory, three are randomly selected. Let $X$ be the number of defective items. Find the probability distribution and variance $D(X)$ of the random variable $X$.

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