“Sumayaw”, “nagsayaw”, and “nasayaw” mean the same thing. In some parts of the Tagalog region, they say “nasayaw” instead of “sumayaw/nagsayaw”. I don't know how “nasayaw” is conjugated though because the few times I heard it, it was used only in the present tense, i.e., in place of “sumasayaw” or “nagsasayaw”. In Manila though, you would very rarely hear people say “nasayaw”. In the song “Doon po sa amin”, I think the word used is “nagsayaw” and not “nasayaw”. It's an old silly song about a group of physically-handicapped people having fun - “nagsayaw ang pilay” (the cripple danced), “kumanta ang pipi” (the mute sang), “nanood ang bulag” (the blind watched), and “nakinig ang bingi” (the deaf listened). The word “magsaing”, when used alone, means “to cook rice”. Its past tense is “nagsaing”. If something else is cooked, it has to be specified. For example, “nagsaing ng isda” would mean, “cooked fish similar to cooking rice”, i.e., with just a small amount of water, similar to giving it the effect of steaming. Hence, in that same song, “nagsaing ng apoy” means “cooked fire”. It also says there, “palayok ay papel” = the crock is (made of) paper; and “tubig na malamig ang ipinanggatong” = cold water was used for fuel. The reason why you got lost with the meanings is because the lyrics don't really make sense and are just meant to be funny. Build a fire is “magsindi ng apoy” (light a fire).

"Sumayaw" is the more proper past tense form, "nasayaw" being a colloquial or dialect-driven version in some parts of the Philippines.

Sumayaw is done in dancing and i think nasayaw is nagsayaw the meaning of nagsayaw is done in dancing its just same meaning