
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 partitioned into $16$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $180^{\circ}$ about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. Calculate the probability the grid is now entirely black.

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