
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 an $8 \times 8$ square, divided into $1 \times 1$ cells. It is covered by right isosceles triangles (two triangles cover one cell). There are 64 black and 64 white triangles. Consider the "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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