
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
Among the $6$ different products mixed together, there are $2$ defective products and $4$ non-defective products. It is required to distinguish them through testing. One product is randomly selected for testing each time, without replacement, until either $2$ defective products or $4$ non-defective products are found.  
$(1)$ If the second product selected is defective and the third product selected is non-defective, how many different ways are there to select the products?  
$(2)$ It is known that the cost of testing each product is $100 dollars. How many ways are there to end the testing with a total cost of $400 dollars? (Note: The solution process should include necessary explanations and steps)

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