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It sounds like your cat has tapeworms. These little worm fragments look like white rice or they might resemble sesame seeds if they are dead.
I suggest that you give your vet a call and get some droncit or drontal plus. Over the counter wormers will not work for tapeworms. The medicine must come from your vet.
Keep in mind that tapeworms come from ingesting fleas so you will want to treat any flea problems you have as well.
Let me know if you still have questions.