Step 1: Understand SN\textsubscript{1 mechanism}
SN\textsubscript{1} is a two-step mechanism involving carbocation formation. Better leaving group and more stable carbocation = faster reaction.
Step 2: Compare leaving groups
\[
\text{Cl}^-<\text{Br}^-<\text{SOCl}_2 (forms intermediate)
\]
Step 3: Reactivity order
Chloride is a better leaving group in polar protic solvents than bromide. Alcohol treated with SOCl\textsubscript{2} is slower due to its mechanism.
\[
\Rightarrow \text{Order: } y>x>z
\]
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