An Equal-Angle Point on a Line

Consider the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$ with right vertex $M(2,0)$. A variable line through the fixed point $P(2,3)$ meets the ellipse at points $A$ and $B$. Prove that there exists a fixed point $N$ on the line $x+2y-2=0$ such that $\angle MNA=\angle MNB$.