
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 cube is made of 27 dice.

a) Find the probability that exactly 25 sixes appear on the surface of the cube.

b) Find the probability that at least one one appears on the surface of the cube.

c) Find the expectation (mathematical expectation) of the number of sixes that are visible on the surface.

d) Find the expectation (mathematical expectation) of the sum of the numbers that appear on the surface of the cube.

e) Find the expectation (mathematical expectation) of the number of distinct digits that appear on the surface of the cube.

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