Paraboloid - Constrained Maxdist

Consider the truncated paraboloid:

$z = x^2 + y^2 \\ 0 \leq z \leq 1$

Let $\vec{P}_1$ and $\vec{P}_2$ be two points on the truncated paraboloid, subject to the following constraint:

$\vec{P}_2 - \vec{P}_1 = \alpha \, \vec{v} \\ \vec{v} = (v_x,v_y,v_z) = (3,1,2) \\ \alpha = \,\, \text{a real number}$

Given the constraints, what is the maximum possible distance between $\vec{P}_1$ and $\vec{P}_2$ ?