Let $\mathcal{P}$ be a regular $2022$-gon with area $1$. Find a real number $c$ such that, if points $A$ and $B$ are chosen independently and uniformly at random on the perimeter of $\mathcal{P}$, then the probability that $AB \geq c$ is $\tfrac{1}{2}$. 

[i]Espen Slettnes[/i]