An Ellipse, a Parabola, and a Circumcircle

As shown in the figure, the ellipse $C_1:\dfrac{x^2}{4}+y^2=1$ and the parabola $C_2:x^2=2py$ (where $p>0$) intersect at two points $A$ and $B$, and $O$ is the origin.

1. If the circumcenter of triangle $ABO$ lies on the ellipse, find the value of $p$.

2. If the circumcircle of triangle $ABO$ passes through the point $N\left(0,\dfrac{13}{2}\right)$, find the value of $p$.