
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  $\mathcal{S}$  be a set of  $2023$  points in a plane, and it is known that the distances of any two different points in  $S$  are all distinct. Ivan colors the points with  $k$  colors such that for every point  $P \in \mathcal{S}$ , the closest and the furthest point from  $P$  in  $\mathcal{S}$  also have the same color as  $P$ . 

What is the maximum possible value of  $k$ ?

*Proposed by Ivan Chan*

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