
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 $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is randomly painted either white or black, with each color equally likely and chosen independently. The square is then rotated $180\,^{\circ}$ about its center. After rotation, any black square in a position formerly occupied by a white square is painted white. The colors of all other squares remain unchanged. Find the probability that the grid is entirely white after this process.

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