Let \(X\) be an octagon. Label its boundary edges, in counterclockwise order, by \[ a,\ b,\ a^{-1},\ b^{-1},\ c,\ d,\ c^{-1},\ d^{-1}, \] where equal labels are identified respecting the indicated orientations. Let \(M\) be the resulting quotient space.
Give a concrete construction showing that \(M\) is homeomorphic to the closed orientable surface of genus \(2\) (the surface of a doughnut with two holes).