
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
Anna and Bob, with Anna starting first, alternately color the integers of the set  $S = \{1, 2, ..., 2022 \}$  red or blue. At their turn each one can color any uncolored number of  $S$  they wish with any color they wish. The game ends when all numbers of  $S$  get colored. Let  $N$  be the number of pairs  $(a, b)$ , where  $a$  and  $b$  are elements of  $S$ , such that  $a$ ,  $b$  have the same color, and  $b - a = 3$ .
Anna wishes to maximize  $N$ . What is the maximum value of  $N$  that she can achieve regardless of how Bob plays?

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