The Mittenpunkt!

Lester's Circle is a triangle circle defined by four triangle centers, the circumcenter, $O$ , the nine-point circle center, $N$ , and the first and second Fermat points $F_1$ and $F_2$ , all of which lie on the circumference of Lester's Circle. In $\triangle ABC$ , the Mittenpunkt , $M$ , also lies on Lester's circle. If the perimeter of $\triangle ABC$ is 56 and the area is 96, what is the radius, $LM$ , of Lester's circle for $\triangle ABC$ ?