
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
In an urn containing a carefully mixed collection of $2n$ balls: $n$ white and $n$ black, what is the probability that each of $n$ people, drawing pairs of balls without looking and without replacement, will draw balls of different colors?

What is the probability that, under the same conditions, each of the $n$ people will draw balls of the same color?

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