
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 factory has three workshops, A, B, and C, which produce the same type of product. The percentage of the total output of each workshop is $25\%$, $35\%$, and $40\%$ respectively. The defective rates of the workshops are $5\%$, $4\%$, and $2\%$ respectively. Now, one product is randomly selected from this batch.<br/>$(1)$ Find the probability of selecting a defective product;<br/>$(2)$ Given that a defective product is selected, find the probability that it was produced in workshop A. (Round to $0.01$).

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