
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
Given a $2 \times 2$ square grid where each square is randomly painted either black or white with equal probability, independently of other squares, find the probability that the entire grid becomes black after the squares at diagonally opposite corners are swapped if a swapped white square moves into a position that was originally black.

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