
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 checkered board of size 2016 by 2017 which is painted alternately with gold and silver paints, numbers 0 or 1 are written at the vertices of each cell such that the sum of the numbers at the vertices of any gold cell is even, and the sum of the numbers at the vertices of any silver cell is odd. What is the possible sum of the numbers written at the four vertices of the entire board?

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