Four identical equilateral-triangle sheets of paper, each with side length $6$, are stacked on top of each other. One sheet is rotated clockwise $30^\circ$ about its center, another by $60^\circ$, and a third by $90^\circ$. The resulting $24$-sided polygon is shown below. Its area can be expressed as $a\sqrt{b}-c$, where $a,b,c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a+b+c$?

1. The outline has $12$-fold rotational symmetry. One $30^\circ$ sector is the union of two triangles, $OPQ$ and $OQT$, where $OP=OT=2\sqrt3$, $OQ=\sqrt6$, and $\angle POQ=\angle QOT=15^\circ$.
2. The area of one sector is $2\left(\frac12\cdot2\sqrt3\cdot\sqrt6\cdot\sin15^\circ\right)=6\sqrt2\sin15^\circ=3(\sqrt3-1)$.
3. The full area is $12\cdot3(\sqrt3-1)=36\sqrt3-36$. Hence $(a,b,c)=(36,3,36)$ and $a+b+c=\boxed{75}$.