A Folded Tetrahedron's Fixed Sphere

In a planar quadrilateral $ABCD$, $AB\perp AD$, $\angle ADC=4\angle ACD=120^\circ$, $CD=2$, and the area of $\triangle ABC$ is $3\sqrt 3$. The triangle $ABD$ is folded along $BD$ to a triangle $PBD$, where $P$ is a moving point.

1. Prove that the volume of the circumscribed sphere of the tetrahedron $P\text{-}BCD$ is a constant. 2. When the distance from $C$ to the plane $PBD$ is $\dfrac{2\sqrt 6}3$, find the cosine of the angle between the lines $PB$ and $CD$.