Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle.  Suppose that $\angle ADE =\angle CBE = 75^\circ$, $BE=4$, and $DE=8$.  The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$.

[i]Proposed by Tony Kim[/i]
1. **Identify Similar Triangles:**
   Given that $\angle ADE = \angle CBE = 75^\circ$, we can infer that $\triangle ADE$ is similar to $\triangle CBE$ by the AA (Angle-Angle) similarity criterion. The ratio of the sides of these triangles is given by:
   \[
   \frac{BE}{DE} = \frac{4}{8} = \frac{1}{2}
   \]
   Therefore, the ratio of the corresponding sides is $\frac{1}{2}$.

2. **Determine Segment Lengths:**
   Since $\triangle ADE \sim \triangle CBE$ and the ratio of similarity is $\frac{1}{2}$, we have:
   \[
   \frac{AE}{EC} = \frac{1}{2}
   \]
   Let $AE = x$ and $EC = 2x$. By the Power of a Point theorem (PoP), we know:
   \[
   AE \cdot EC = BE \cdot DE
   \]
   Substituting the known values:
   \[
   x \cdot 2x = 4 \cdot 8 \implies 2x^2 = 32 \implies x^2 = 16 \implies x = 4
   \]
   Thus, $AE = 4$ and $EC = 8$.

3. **Determine Angles:**
   Since $\triangle ADE$ is isosceles with $AE = DE = 4$, the base angles are equal. Therefore:
   \[
   \angle AED = 180^\circ - 2 \times 75^\circ = 30^\circ
   \]
   Consequently, $\angle AEB = 180^\circ - \angle AED = 150^\circ$.

4. **Apply Law of Cosines:**
   To find $AB^2$, we use the Law of Cosines in $\triangle AEB$:
   \[
   AB^2 = AE^2 + BE^2 - 2 \cdot AE \cdot BE \cdot \cos(\angle AEB)
   \]
   Substituting the known values:
   \[
   AB^2 = 4^2 + 8^2 - 2 \cdot 4 \cdot 8 \cdot \cos(150^\circ)
   \]
   Since $\cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}$, we have:
   \[
   AB^2 = 16 + 64 - 2 \cdot 4 \cdot 8 \cdot \left(-\frac{\sqrt{3}}{2}\right)
   \]
   Simplifying further:
   \[
   AB^2 = 16 + 64 + 32\sqrt{3} = 80 + 32\sqrt{3}
   \]

5. **Identify Constants:**
   The value of $AB^2$ is in the form $a + b\sqrt{c}$, where $a = 80$, $b = 32$, and $c = 3$. Therefore:
   \[
   a + b + c = 80 + 32 + 3 = 115
   \]

The final answer is $\boxed{115}$