3D Kissing Number Challenge (Medium)

Our Specific Setup:

• Dimension: 3
• Central Sphere Radius (R): 3.14
• Satellite Sphere Radius (r): 1.0

Rules:

1. No Crashing: The distance between any two satellite centers must be at least 2.0 (2 * r).
2. Touching the Center: Centers should be at distance 4.14 (R + r) from the origin.
   (Note: If your vectors point in the right direction but have the wrong length, our system will automatically renormalize them for you!)

Your Goal:

Find the longest possible list of satellite sphere centers. The longer the list, the better! (When you think the number spheres has reached the maximum, it might be a fun activity to prove your conjecture.)