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Is it a 2002 Daewoo Lanos?
The normal recommended oil viscosity for this engine is 10w30 for most climates. In colder climates or in winter then 5w30 is recommended, in extreme hot conditions then 15w40 is recommended. I'm not sure of your location, but most places in the US I would recommend 10w30 for most of the year, possibly switching to 5w30 in the winter.
Synthetic or conventional oil is just up to preference in this case, the manufacturer did not specify one way or the other.
If the oil is low and you're unsure of the history then it wouldn't hurt to just have the oil changed and start fresh, that way you would know when it was last done and what oil is in it.
as I understand that once you used the synthetic oil in your engine than you can not switch back to conventional oil. is this correct.
is it going to hurt anything if you mix synthetic oil to conventional oil to a engine, or 10-20 to 20-40 grade oil to the same engine?
No, it won't really hurt anything if you mix the oil, but if you don't know what type or viscosity the oil is now then it would be a good idea to just do an oil change and start fresh.