
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 an $8 \times 8$ board where the cells have not been colored yet. In how many ways can the board be colored in black and white such that there are 31 black cells and no two black cells share a common side? (Two coloring methods are considered different if there is at least one cell that is white in one coloring and black in another.)

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