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Yes you can use rigid to pipe your vent. I fact that's what's typically used for longer vent runs.
Without a model number for your unit and venting specs there's no way I can say for sure. But 12' isn't very far and if there's only a couple elbows in the run you should be OK.
Do you have any more questions?
Specs call for 6" round minimum size vent so 4" doesn't meet spec and your 12' run won't make much of a difference