
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
In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from   $1$  to  $n$  and let  $a_i$  be the number of friends of the  $i^{\text{th}}$  resident. Suppose that
\[ \sum_{i=1}^{n}a_i^2=n^2-n \]
Let  $k$  be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of  $k.$ 

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