
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
8. Shortening. There is a sequence of 2015 digits. All digits are chosen randomly from the set {0, 9} independently of each other. The following operation is performed on the resulting sequence. If several identical digits go in a row, they are replaced by one such digit. For example, if there was a fragment ...044566667..., then it becomes ...04567...

a) Find the probability that the sequence will shorten by exactly one digit.

b) Find the expected length of the new sequence.

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