A Vertex-Triangle Fixed Point

The ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has left and right vertices $A, B$ and top vertex $C$, eccentricity $\dfrac{\sqrt{3}}{2}$, and the triangle $ABC$ has area $2$.

1. Find the standard equation of the ellipse $E$. 2. Let $D$ be a moving point of $E$ in the first quadrant, distinct from the vertices. Line $BD$ meets line $AC$ at $M$, and line $CD$ meets the $x$-axis at $N$. Prove that the line $MN$ passes through a fixed point.