Before revision:  By ``hopping'' iteratively to such larger cycles, we obtain a powerful and very general inductive motor for proving and computing long cycles (we will give an algorithm with running time O(n^2)). This provides a method to prove  lower bounds on  Tutte cycles, as C' will be a Tutte cycle of G if C is.
After revision:  By ``hopping'' iteratively to such larger cycles, we obtain a powerful and very general inductive motor for proving and computing long cycles (we will give an algorithm with quadratic running time). This is the first step towards the so far elusive quest of finding a universal induction that captures longest cycles of polyhedral graph classes. Our motor provides also a method to prove lower bounds on Tutte cycles, as C' will be a Tutte cycle of G if C is.