Cycle of generalized eigenvectors disjoint

Question: Let be cycles of generalized eigenvectors of a linear operator with respect to eigenvalue . Prove that if the initial eigenvectors are distinct, then the cycles are disjoint.

Attempt: Suppose that are (distinct) generalized eigenvectors of and let be the smallest integer such that and .

I tried to argue by contradiction. Say that , then there exist and such that . Then for all and , we have . And then, this needs to imply that are in fact not distinct anymore and thus creates a contradiction ...?