
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
Given that there are 2 defective products and 3 non-defective products mixed together, we need to identify them through testing. We test one product at a time, without returning the product after testing, until either 2 defective products are identified or 3 non-defective products are found.  
(1) Determine the probability that the first test identifies a defective product and the second test identifies a non-defective product.  
(2) Assume that each test costs 100 yuan. Let $X$ denote the total cost of testing until 2 defective products or 3 non-defective products are identified (in yuan). Find the probability distribution and mathematical expectation of $X$.

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