Let \( ABC\) be an equilateral triangle in the plane with side length of \(1\) with vertices labeled in counterclockwise order. Let point \(A'\) coincide with \(A\), \(B'\) coincide with \(B\), and \(C'\) coincide with \(C\). Suppose the leg \(AB'\) rotates about \(A\) counterclockwise, \(CA'\) rotates about \(C\) counterclockwise, and \(BC'\) rotates about \(B\) counterclockwise all at the same rate until the three legs intersect at one point. Each leg intersects another leg at a point. These \(3\) intersection points trace out paths as the legs rotate. Find the total length of these paths.
Note: The diagram is not completely accurate or to scale.