\noindent\textbf{1} \hfill \textit{Go to the solution page}

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\noindent Let $\mathrm{A}(0,\ -1,\ 1)$ be a point in coordinate space. Suppose a point $\mathrm{P}$ in the $xy$-plane satisfies all of the following conditions (i), (ii), (iii).

\begin{itemize}
  \item[(i)] $\mathrm{P}$ is different from the origin $\mathrm{O}$.
  \item[(ii)] $\angle \mathrm{AOP} \geq \dfrac{2}{3}\pi$
  \item[(iii)] $\angle \mathrm{OAP} \leq \dfrac{\pi}{6}$
\end{itemize}

\noindent Sketch the region that $\mathrm{P}$ can occupy in the $xy$-plane.



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