
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
Alice writes  $1001$  letters on a blackboard, each one chosen independently and uniformly at random from the set  $S=\{a, b, c\}$ . A move consists of erasing two distinct letters from the board and replacing them with the third letter in  $S$ . What is the probability that Alice can perform a sequence of moves which results in one letter remaining on the blackboard?

*Proposed by Daniel Zhu.*

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