Fundamental group of the crown with quotient?

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I am studying now the Seifert-Van-Kampen's theorem and I am trying to find some other examples than torus, Klein bottle, etc... that seem easy with this theorem.

I had an example of a quotient space which is the circular crown with the quotient that identifies antipodal points in each circle.

I've been trying to find a partition in this space to aply the theorem in a similar way of torus and klein bottle but I don't know what are the fundamental groups of the spaces I get.

Is there any way to find this fundamental group with this theorem?