
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
2021 unit squares are arranged in a row, with the leftmost and rightmost squares both colored black, and the remaining 2019 squares are white. Two adjacent white squares are randomly selected, and one of them is randomly dyed black. This process is repeated until there are no adjacent white squares left. Let \( E \) be the expected number of remaining white squares. Find the greatest integer not exceeding \( E \).

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