
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
Peter and Paul are playing with marbles. Peter has 12 white and 8 black marbles, and Paul has 8 white and 12 black marbles, all of which they keep in urns. Step by step, each of them simultaneously (without looking) transfers a marble from their own urn to the other's. What is the probability that on the \(k\)-th move, Peter transfers a white marble to Paul and Paul transfers a black marble to Peter?

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