Given a cone C C defined by the equation z 2

( x − 2 ) 2 + ( y + 3 ) 2 z 2 =(x−2) 2 +(y+3) 2 that extends infinitely in the z-direction, and a sphere S S defined by ( x − 1 ) 2 + ( y + 1 ) 2 + ( z − 4 ) 2

36 (x−1) 2 +(y+1) 2 +(z−4) 2 =36, analyze the spatial relationship between the cone and the sphere by determining the conditions for intersection, including cases where the cone intersects the sphere along a curve, touches the sphere tangentially, or lies completely outside the sphere. Furthermore, calculate the volume of the solid formed by the intersection when applicable.

What is the final answer to the given problem? *

Determine the correct mathematical answer for the prompt above. The final answer should be in the simplest form (e.g., only include the answer; complete sentences are unnecessary; this should rarely be longer than a single line).