Three identical square sheets of paper each with side length 6 are stacked on top of each other. The middle sheet is rotated clockwise $30^{\circ}$ about its center and the top sheet is rotated clockwise $60^{\circ}$ about its center, resulting in the $24$ -sided polygon shown in the figure below.

The area of this polygon can be expressed in the form $a-b \sqrt{c}$, where $a, b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime.
What is $a+b+c$ ?