
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  $n$  be a given positive integer. Say that a set  $K$  of points with integer coordinates in the plane is connected if for every pair of points  $R, S\in K$ , there exists a positive integer  $\ell$  and a sequence  $R=T_0,T_1, T_2,\ldots ,T_{\ell}=S$  of points in  $K$ , where each  $T_i$  is distance  $1$  away from  $T_{i+1}$ . For such a set  $K$ , we define the set of vectors 
\[\Delta(K)=\{\overrightarrow{RS}\mid R, S\in K\}\]
What is the maximum value of  $|\Delta(K)|$  over all connected sets  $K$  of  $2n+1$  points with integer coordinates in the plane?

*Grigory Chelnokov, Russia*

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