Shortest Distance from Circle to Parabola (Part 2)

In the $xyz$ coordinate system, there is a circle with the following properties:

Center: $\vec{C} = (0,0,2)$
Normal Vector to Plane Containing Circle: $\vec{N} = (1,1,1)$
Radius: $R = 5$

The is also a parabola with the following parametrization:

$\large{\vec{P} = \alpha \, \vec{u_1} + \alpha^2 \, \vec{u_2}}$

In the above expression, $\vec{u_1}$ is a unit-vector in the direction of $(-1,2,3)$ and $\vec{u_2}$ is a unit-vector in the direction of $(2,-5,4)$ . The variable $\alpha$ is a spatial parameter.

What is the minimum distance from the circle to the parabola?