
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 that there are 2017 non-overlapping circles on the blackboard, with no three circles having a common tangent line. Define a tangent segment between two circles as the line segment connecting the two points of tangency of a common tangent line. Luciano draws tangent segments on the blackboard one by one, ensuring that each tangent segment does not intersect any other circles or previously drawn tangent segments. He continues drawing tangent segments according to these rules until he cannot draw any more. Determine all possible numbers of tangent segments that Luciano could have drawn when he stops.

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