
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
A sequence  $P_1, \dots, P_n$  of points in the plane (not necessarily diferent) is *carioca* if there exists a permutation  $a_1, \dots, a_n$  of the numbers  $1, \dots, n$  for which the segments  $$ P_{a_1}P_{a_2}, P_{a_2}P_{a_3}, \dots, P_{a_n}P_{a_1} $$  are all of the same length.

Determine the greatest number  $k$  such that for any sequence of  $k$  points in the plane,  $2023-k$  points can be added so that the sequence of  $2023$  points is *carioca*.

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