
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 has three workshops, A, B, and C, producing the same product. It is known that the output of each workshop accounts for $25\%$, $35\%$, and $40\%$ of the total output of the factory, respectively. The defect rates of the workshops are $5\%$, $4\%$, and $2%$, respectively. Now, one item is randomly selected from the products of this factory.<br/>$(1)$ Find the probability of selecting a defective item;<br/>$(2)$ If a defective item is selected, what are the probabilities that it was produced by each of the three workshops?

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