
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
Let there be a regular polygon of  $n$  sides with center  $O$ . Determine the highest possible number of vertices  $k$   $(k \geq 3)$ , which can be coloured in green, such that  $O$  is strictly outside of any triangle with  $3$  vertices coloured green. Determine this  $k$  for  $a)  n=2019$  ;   $b)  n=2020$ .

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