Hatcher exercise 1.2.13

Question

In this exercise we consider a disk with two holes and we identify the three boundary circles. There are only two essentially different ways of identifying the three boundary circles: one gives a modified Klein bottle, say $Y$, and the other gives another topological space $Z$. I have to show that their fundamental groups are different (i.e $\pi_1(Y)$ is not isomorphic to $\pi_1(Z)$). My problem is: I have no idea how to compute these fundamental groups. Any suggestion of method ?