You always land in "your" cycle, because you start with the box with your name on it.

All sequences of box-opening that use the method described must eventually cycle, because both box-to-name mappings are 1-to-1. Because it's a cycle, the sequence must eventually lead back to the box you started with. Since you start with the box with your name on it, then whatever cycle you landed it, it definitely contains the box with your name on it, because that box is the one that completes the cycle. The only question is whether the cycle is length-50 or less.