
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 constructed from 27 dice.

a) Find the probability that exactly 25 of the faces showing on the surface of the cube have the number six.

b) Find the probability that at least one of the faces showing on the surface of the cube has the number one.

c) Find the expected number of faces showing the number six on the surface of the cube.

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

e) Find the expected value of the random variable: "The number of different digits appearing on the surfaces of the dice."

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