
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
Every positive integer is marked with a number from the set  $\{ 0,1,2\}$ , according to the following rule:  $$ \text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0. $$  Let  $S$  denote the sum of marks of the first  $2019$  positive integers. Determine the maximum possible value of  $S$ .

*Proposed by Ivan Novak*

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