
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 a box, there are \( n \) white and \( n \) black balls. Pairs of balls are drawn randomly from the box two at a time.

a) Find the expected number of mixed-color pairs that are drawn from the box by the time the box is empty.

b) Suppose that if a pair is mixed-color, it is set aside, and if it is same-color, it is returned to the box. We call this operation an "attempt." Find the expected number of attempts needed such that all balls are set aside and the box is empty.

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