
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
A board of size \(2022 \times 2022\) is given. Lisa and Varya take turns painting \(2 \times 2\) squares on the board with red and blue colors. They agreed that each cell can be painted no more than once in blue and no more than once in red. Cells that are painted blue and then red (or vice versa) become purple. Once all cells are painted, the girls count how many of them are purple. What counts could they have gotten?

Options:
- \(2022 \times 2022\)
- \(2022 \times 2020\)
- \(2021 \times 2022\)
- \(2021 \times 2020\)

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