
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 one of the planets of Tau Ceti, among older women, the following activity is common: a grid board of size 2016 by 2017 is painted with gold and silver paints in a checkerboard pattern. Then, in the vertices of each cell, the numbers 0 or 1 are written in such a way that the sum of the numbers at the vertices of any golden cell is even, and the sum of the numbers at the vertices of any silver cell is odd. What could be the 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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