
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 simple graph  $G$ , we call  $t$  pairwise adjacent vertices a  $t$ *-clique*. If a vertex is connected with all other vertices in the graph, we call it a *central* vertex. Given are two integers  $n,k$  such that  $\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n$ . Let  $G$  be a graph on  $n$  vertices such that**(1)**  $G$  does not contain a  $(k+1)$ -*clique*;**(2)** if we add an arbitrary edge to  $G$ , that creates a  $(k+1)$ -*clique*.
Find the least possible number of *central* vertices in  $G$ .

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