An ellipse has foci A and B and has the property that there is some point C on the ellipse such that the area of the circle passing through A, B, and, C is equal to the area of the ellipse. Let e be the largest possible eccentricity of the ellipse. One may write e^{2} as \frac{a+\sqrt{b}}{c}, where a, b, and c are integers such that a and c are relatively prime, and b is not divisible by the square of any prime. Find a^{2}+b^{2}+c^{2}.