
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
Determine all integers  $n$  for which there exist an integer  $k\geq 2$  and positive integers  $x_1,x_2,\hdots,x_k$  so that  $$ x_1x_2+x_2x_3+\hdots+x_{k-1}x_k=n\text{ and } x_1+x_2+\hdots+x_k=2019. $$  

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