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Hello. Thanks for the question.
In a young cat with lethargy, I would guess you're dealing with a fever. The laying on cold tiles also fits with a fever. Without any upper respiratory signs (sneezing, ocular discharge), it's most likely Calicivirus. This is a common virus that can cause a fever and joint pain. These cats/ kittens are lethargic and reluctant to move around much.
Most will resolve on their own over a few days. I do like to use an antibiotic in case of any secondary bacterial infection. I also like to use Onsior or Metacam- these are anti-inflammatories that help to get the fever down faster. You could get those from a vet.
I hope this helps, let me know if you have any other questions.
I'm sorry for that wait. I got called away for an emergency. The cause is a virus. We usually don't know how they get exposed, as it's a common virus in the environment.