
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 manufactures a certain type of product; each product has a defect with a probability of $p_{1}=0,1$. The product is inspected by one controller; he detects an existing defect with a probability of $p_{2}=0,8$, and if the defect is not detected, he passes the product into the finished goods. Additionally, the controller may mistakenly reject a non-defective product with a probability of $p_{3}=0,3$. Find the probabilities of the following events:

$A_{1}$ - the product will be mistakenly rejected;

$A_{2}$ - the product will be passed into the finished goods with a defect;

$A_{3}$ - the product will be rejected.

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