
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 cubical dice tower is constructed with three cubes stacked on top of each other. Each face of each cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge with the choice of the edge pairing made at random and independently for each face. Calculate the probability that there is a continuous stripe running vertically from the top cube, through the middle cube, to the bottom cube across at least one of their shared vertical faces.

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