
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 $4 \times 4$ square is divided into $16$ unit squares. Each unit square is painted either white or black, each with a probability of $\frac{1}{2}$, independently. The square is then rotated $180^\circ$ about its center. After rotation, any white square that occupies a position previously held by a black square is repainted black; other squares retain their original color. Determine the probability that the entire grid is black after this process.

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