*First Morley Triangles*. Observe the following image:

The triangle shown above is a 45-45-90 triangle with a hypotenuse of length 4. If the area of the First Morley Triangle depicted in this image can be written as $a\sqrt{b}-c$, where $a$, $b$, and $c$ are positive integers and $b$ is square-free, what is $a+b+c$?