
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
Pensioners on one of the planets in Alpha Centauri enjoy painting the cells of $2016 \times 2016$ boards with gold and silver paints. One day, it turned out that for all the boards painted that day, each $3 \times 3$ square contained exactly $A$ gold cells, and each $2 \times 4$ or $4 \times 2$ rectangle contained exactly $Z$ gold cells. For which values of $A$ and $Z$ is this possible?

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