Addiction is a scary thing
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What would happen if we put a small collider inside of a bigger collider and spun it around while it spun around?
"Yo dawg, I heard you like colliders, so we built a collider into your collider so it can collide while it collides...."
SOUNDS LIKE A GOOD QUESTION 22 BILLION DOLLARS WOULD ANSWER.
You are asking something different, but I think it's interesting to mention that the particles that go into the LHC don't start there. The LHC gets them from the SPS, which gets them from the PS and this keeps going for a few more steps.
https://www.smbc-comics.com/index.php?db=comics&id=3554
We recreate an atom?