Consider the curve $C: y = x^3 - x$ in the coordinate plane.
[(1)] Show that every point $\mathrm{P}$ in the coordinate plane satisfies the following condition (i).
[(i)] There exists a line $l$ passing through $\mathrm{P}$ that intersects the curve $C$ at 3 distinct points.
[(2)] Sketch the region of all possible positions of point $\mathrm{P}$ in the coordinate plane that satisfy the following condition (ii).
[(ii)] There exists a line $l$ passing through $\mathrm{P}$ that intersects the curve $C$ at 3 distinct points, and moreover, the two regions enclosed by the line $l$ and the curve $C$ have equal areas.
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Consider the curve $C: y = x^3 - x$ in the coordinate plane.
\begin{enumerate}
\item[(1)] Show that every point $\mathrm{P}$ in the coordinate plane satisfies the following condition (i).
\begin{enumerate}
\item[(i)] There exists a line $l$ passing through $\mathrm{P}$ that intersects the curve $C$ at 3 distinct points.
\end{enumerate}
\item[(2)] Sketch the region of all possible positions of point $\mathrm{P}$ in the coordinate plane that satisfy the following condition (ii).
\begin{enumerate}
\item[(ii)] There exists a line $l$ passing through $\mathrm{P}$ that intersects the curve $C$ at 3 distinct points, and moreover, the two regions enclosed by the line $l$ and the curve $C$ have equal areas.
\end{enumerate}
\end{enumerate}
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