
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 home service company classifies its housekeepers into two categories based on customer satisfaction: there are 12 category \\(A\\) housekeepers and \\(x\\) category \\(B\\) housekeepers.

\\((\\)I\\()\\) If stratified sampling is used to randomly select 20 housekeepers for technical training, and 16 of them are category \\(B\\) housekeepers, find the value of \\(x\\);

\\((\\)II\\()\\) A customer comes to the company to hire 2 housekeepers, but due to the company's personnel arrangement being nearly saturated, there are only 3 category \\(A\\) housekeepers and 2 category \\(B\\) housekeepers to choose from.

\\((ⅰ)\\) Please list all possible choices for this customer;

\\((ⅱ)\\) Calculate the probability that the customer ultimately hires housekeepers from both category \\(A\\) and category \\(B\\).

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