
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
Consider a \(4 \times 4\) grid where each unit square is independently colored either red or blue with equal likelihoods. The grid is then rotated 180 degrees about its center, and any previously red square that is now in a position of a blue square will turn blue. The process (including the random coloring and the rotation) is then repeated once more. What is the probability that the entire grid becomes blue after these operations?
A) \(\frac{1}{256}\)
B) \(\frac{1}{2048}\)
C) \(\frac{1}{512}\)
D) \(\frac{1}{4096}\)
E) \(\frac{1}{65536}\)

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