
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
Given a graph with  $n$  ( $n\ge 4$ ) vertices . It is known that for any two vertices  $A$  and  $B$  there exists a vertex which is connected by edges both with  $A$  and  $B$ . Find the smallest possible numbers of edges in the graph. 

E. Barabanov


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