MathNet 2014 Question 16

Let $\mathcal{K}$ be a circle with centre $O$ and radius $r$. What is the upper bound for the length of $|OT|$ given that $T$ is the point in the plane such that there exists an equilateral triangle $ABT$, where $A$ and $B$ lie on the circle $\mathcal{K}$?



Supplemental schematic, not to scale.