
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
Each face of a cube is painted black, white, or gray, each with equal probability. Each color choice per face is independent of the other faces. What is the probability that the cube can be positioned on a horizontal surface so that the four vertical faces are all the same color, and each remaining top and bottom face matches (either both black, both white, or both gray)?

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