Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $22.5^{\circ }$ about its center and the top sheet is rotated clockwise $45^{\circ }$ about its center, resulting in a $16$-sided polygon. 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 $a+b+c$?