Satisfying assignments, twice-3SAT NP complete I wanted to solve the following problem about 3SAT . The question is 
1. to show if the problem is NP-complete and 
2. whether the problem has two different satisfying assignments.

"TWICE-3SAT Input: A propositional formula ϕ in conjunctive normal form, such that each clause consists of exactly three literals (as in 3SAT). Question: Does ϕ have at least two different satisfying assignments?" 

I understand that we have to use reduction of a known NP-complete problem (such as an independent set) to the problem asked. But I can't go further. I would appreciate your help. I have been working on this for almost a week but I cannot go further.
Hints:


Showing that the problem is NP is easy.
To show that it is NP-hard, show first that TWICE-SAT is NP-hard (see next step).
In order to show that TWICE-SAT is NP-hard, reduce from SAT. The idea is to add a "dummy" satisfying assignment, which is always going to be satisfying; any satisfying assignment of the original SAT instance will translate to a (different) satisfying assignment of the new instance.
Either reduce TWICE-SAT to TWICE-3SAT, or use insights from your proof in step 3 to prove that TWICE-3SAT is NP-hard.