
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 large cube is composed of 27 dice.
a) Find the probability that exactly 25 sixes are showing on the surface of the large cube.

b) Find the probability that at least one 'one' is showing on the surface of the large cube.

c) Determine the expected number of sixes showing on the surface.

d) Calculate the expected sum of the numbers showing on the surface of the cube.

e) Find the expected number of distinct digits appearing on the surface of the cube.

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