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Dose as a noun means a measured amount of something, especially medicine. As a verb it means to give someone a measured amount of medicine.

For example: Even with the medicine we dose him with, he seems to be getting worse instead of better.

Doze is also a noun and a verb. The verb means to have a short sleep, especially during the day and "a doze" is a short sleep.

For example: I often doze off after dinner.