
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 are on the surface of the cube.

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

c) Find the expected number of sixes facing outward.

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

e) Find the expected value of the random variable: "The number of different digits that appear on the surface of the cube."

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