A Mixed Conic Fixed-Point Chord

Consider the ellipse $\dfrac{x^2}5+\dfrac{y^2}4=1$ and the circle $x^2+y^2=1$. Let $P$ be a moving point on the ellipse. A point $A$ on the ellipse is such that the line $AP$ passes through $M(0,1)$, and a point $B$ on the circle is such that the line $BP$ passes through $N\left(0,\dfrac 12\right)$, with $B$ and $P$ on opposite sides of the $y$-axis. Prove that the line $AB$ passes through a fixed point.