
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 \(2^{2014}+1\) by \(2^{2014}+1\) grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer \(n\) greater than 2, there do not exist pairwise distinct black squares \(s_{1}, s_{2}, \ldots, s_{n}\) such that \(s_{i}\) and \(s_{i+1}\) share an edge for \(i=1,2, \ldots, n\) (here \(s_{n+1}=s_{1}\) ).

What is the maximum possible number of filled black squares?

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