
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 simple graph with  $ 4038 $  vertices. Assume we arbitrarily choose  $ 2019 $  vertices as a group (the other  $ 2019 $  is another group, of course), there are always  $ k $  edges that connect two groups. Find all possible value of  $  k $ .

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