
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
Suppose you have  $27$  identical unit cubes colored such that  $3$  faces adjacent to a vertex are red and the other  $3$  are colored blue. Suppose further that you assemble these  $27$  cubes randomly into a larger cube with  $3$  cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as  $\frac{1}{2^n}$  for some positive integer  $n$ . Find  $n$ .

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