
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
There are $10$ different products, of which $3$ are defective.

$(1)$ If $3$ products are randomly selected for inspection, how many possible selections contain at least one defective product?

$(2)$ If each product is tested one by one until all $3$ defective products are found, and it turns out that the third defective product is found exactly on the fifth test, how many different testing methods are there?

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