
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 $4$ workshops producing the same type of computer component. The output of the $4$ workshops accounts for $15\%$, $20\%$, $30\%$, and $35\%$ of the total output, respectively. It is known that the defect rates of these $4$ workshops are $0.04$, $0.03$, $0.02$, and $0.01$ respectively. Now, if one piece of this product is randomly selected from the factory, find:<br/>$(1)$ The probability of selecting a defective product exactly?<br/>$(2)$ Given that a selected product is defective, what is the probability that it came from the second workshop?

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