Find the product of all possible real values for $k$ such that the system of equations
$$x^2+y^2= 80$$
$$x^2+y^2= k+2x-8y$$

has exactly one real solution $(x,y)$.

[i]Proposed by Nathan Xiong[/i]
1. **Identify the given equations and their geometric interpretations:**
   The given system of equations is:
   \[
   x^2 + y^2 = 80
   \]
   \[
   x^2 + y^2 = k + 2x - 8y
   \]
   The first equation represents a circle \(\Gamma\) centered at the origin \((0,0)\) with radius \(\sqrt{80}\).

2. **Rearrange and complete the square for the second equation:**
   \[
   x^2 + y^2 = k + 2x - 8y
   \]
   Rearrange terms:
   \[
   x^2 - 2x + 1 + y^2 + 8y + 16 = k + 17
   \]
   This simplifies to:
   \[
   (x-1)^2 + (y+4)^2 = k + 17
   \]
   This represents a circle \(\omega\) centered at \((1, -4)\) with radius \(\sqrt{k + 17}\).

3. **Determine the conditions for tangency:**
   For the system to have exactly one real solution, the circles \(\Gamma\) and \(\omega\) must be tangent to each other. The distance between the centers of the circles is:
   \[
   \text{Distance} = \sqrt{(1-0)^2 + (-4-0)^2} = \sqrt{1 + 16} = \sqrt{17}
   \]
   The radius of \(\Gamma\) is \(\sqrt{80}\). Let \(r = \sqrt{k + 17}\) be the radius of \(\omega\).

4. **Set up the tangency condition:**
   The circles are tangent if the sum or difference of their radii equals the distance between their centers:
   \[
   \sqrt{80} + \sqrt{k + 17} = \sqrt{17} \quad \text{or} \quad \sqrt{80} - \sqrt{k + 17} = \sqrt{17}
   \]

5. **Solve for \(k\):**
   - Case 1: \(\sqrt{80} + \sqrt{k + 17} = \sqrt{17}\)
     \[
     \sqrt{k + 17} = \sqrt{17} - \sqrt{80}
     \]
     Square both sides:
     \[
     k + 17 = (\sqrt{17} - \sqrt{80})^2
     \]
     \[
     k + 17 = 17 + 80 - 2\sqrt{17 \cdot 80}
     \]
     \[
     k + 17 = 97 - 2\sqrt{1360}
     \]
     \[
     k = 97 - 2\sqrt{1360} - 17
     \]
     \[
     k = 80 - 2\sqrt{1360}
     \]

   - Case 2: \(\sqrt{80} - \sqrt{k + 17} = \sqrt{17}\)
     \[
     \sqrt{k + 17} = \sqrt{80} - \sqrt{17}
     \]
     Square both sides:
     \[
     k + 17 = (\sqrt{80} - \sqrt{17})^2
     \]
     \[
     k + 17 = 80 + 17 - 2\sqrt{80 \cdot 17}
     \]
     \[
     k + 17 = 97 - 2\sqrt{1360}
     \]
     \[
     k = 97 - 2\sqrt{1360} - 17
     \]
     \[
     k = 80 - 2\sqrt{1360}
     \]

6. **Calculate the product of all possible \(k\):**
   Since both cases yield the same value for \(k\), the product of all possible \(k\) is:
   \[
   k_1 = 80 - 2\sqrt{1360}
   \]
   \[
   k_2 = 80 - 2\sqrt{1360}
   \]
   \[
   k_1 \cdot k_2 = (80 - 2\sqrt{1360}) \cdot (80 - 2\sqrt{1360}) = (80 - 2\sqrt{1360})^2
   \]
   \[
   k_1 \cdot k_2 = 80^2 - 4 \cdot 80 \cdot \sqrt{1360} + 4 \cdot 1360
   \]
   \[
   k_1 \cdot k_2 = 6400 - 320 \cdot \sqrt{1360} + 5440
   \]
   \[
   k_1 \cdot k_2 = 11840 - 320 \cdot \sqrt{1360}
   \]

The final answer is \(\boxed{960}\)