An example of non-homeomorphic surfaces, $S_1, S_2$, such that $S_1 \times [0,1]$ is homeomorphic to $S_2 \times [0,1]$

Question

I'm just reviewing the questions I couldn't answer in an exam I gave for my Topology course, one of the questions asked me the following:

To give an example of two compact connected surfaces $S_1, S_2$, possibly with boundary, such that $S_1$ is not homeomorphic to $S_2$, but $S_1 \times [0,1]$ is homeomorphic to $S_2 \times [0,1]$.

I would be grateful if someone could give me an answer, or even an approach to this problem, because I'm drawing a complete blank