
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
On a plane, there is a square with dimensions \(8 \times 8\), divided into \(1 \times 1\) squares. It is covered with isosceles right triangles (two triangles cover one square). There are 64 black and 64 white triangles available. Consider "correct" coverings, such that any two triangles sharing a side are of different colors. How many correct coverings exist?

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