Let \(n\) be a real number; define the curve \(C_n\) by the equation
\[x^n + y^n = 1.\ \ \ \ x, y \geq 0\]
For every point \(x, y\) on \(C_n\) there is a tangent line. This tangent line intersects both coordinate axes. Consider the line segment between these intersection points.
There exists one curve \(C_n\) for which all these tangential line segments have the same length. How much is \(n\)?