
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 larger cube has been constructed from 27 smaller dice. 

a) Find the probability that exactly 25 of the outer faces of the larger cube show a six.

b) Find the probability that there is at least one face showing a one on the outer surface of the larger cube.

c) Find the expected number of sixes showing on the outer surface of the larger cube.

d) Find the expected sum of the numbers on the outer surface of the larger cube.

e) Find the expected number of different digits showing on the outer surface of the larger cube.

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