
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
Given a $2022 \times 2022$ board, Olya and Masha take turns to color $2 \times 2$ squares on it using red and blue. They agreed that each cell can be painted no more than once in blue and no more than once in red. Cells 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 possible counts could they have?

### Options
○ $2022 \cdot 2022$

○ $2022 \cdot 2021$

○ $2022 \cdot 2020$

○ $2020 \cdot 2020$

### Correct Answers
○ $2022 \cdot 2020$

○ $2020 \cdot 2020$

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