(12 points) Let $O$ be the origin of coordinates. Point $M$ is on the ellipse $C: \dfrac{x^2}{2} + y^2 = 1$. The perpendicular from $M$ to the $x$-axis intersects the $x$-axis at $N$. Point $P$ satisfies $\overrightarrow{NP} = \sqrt{2}\,\overrightarrow{NM}$. (1) Find the trajectory equation of point $P$. (2) Let point $Q$ be on the line $x = -3$, and $\overrightarrow{OP} \cdot \overrightarrow{PQ} = 1$. Prove that the line $l$ passing through point $P$ and perpendicular to $OQ$ passes through the right focus $F$ of $C$.
(12 points)
Let $O$ be the origin of coordinates. Point $M$ is on the ellipse $C: \dfrac{x^2}{2} + y^2 = 1$. The perpendicular from $M$ to the $x$-axis intersects the $x$-axis at $N$. Point $P$ satisfies $\overrightarrow{NP} = \sqrt{2}\,\overrightarrow{NM}$.
(1) Find the trajectory equation of point $P$.
(2) Let point $Q$ be on the line $x = -3$, and $\overrightarrow{OP} \cdot \overrightarrow{PQ} = 1$. Prove that the line $l$ passing through point $P$ and perpendicular to $OQ$ passes through the right focus $F$ of $C$.