
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
Consider a horizontal strip of  $N+2$  squares in which the first and the last square are black and the remaining  $N$  squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color tis neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let  $w(N)$  be the expected number of white squares remaining. Find
\[ \lim_{N\to\infty}\frac{w(N)}{N}.\]

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