= Solution
Every iterate $g^n(x,y)=(x+n,(-1)^n y)$ is an <isometry> of the strip with its Euclidean subspace metric. Let $a,b$ lie in distinct <orbits>. Since the first-coordinate difference in $a-g^n b$ grows without bound as $|n|\to\infty$, the positive distances $d(a,g^n b)$ have a positive minimum $\delta$. Only finitely many indices can give a <distance> below any fixed bound, and none gives zero.
Choose $0<\varepsilon<\delta/3$. Saturate the relative <open balls> by setting
$$
U=\bigcup_{n\in\mathbb Z}g^nB_S(a,\varepsilon),\qquad
V=\bigcup_{n\in\mathbb Z}g^nB_S(b,\varepsilon).
$$
These are <open> saturated <neighbourhoods> of the two <orbits>. Any two centres $g^n a,g^m b$ are at least $\delta$ apart, since their <distance> equals $d(a,g^{m-n}b)$. Their radius-$\varepsilon$ balls therefore cannot intersect. Hence $U\cap V=\varnothing$, and part (a)(iv), an instance of <positive orbit separation implies a Hausdorff quotient>, proves
$$
\boxed{S/\mathcal R'\text{ is Hausdorff}.}
$$
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