If x ∈ {0, 1}ⁿ is a boolean string, let x ∈ {0, 1}ⁿ denote the new string obtained by flipping the bit in every coordinate in x. That is, xi := 1 − xi for every i ∈ [n]. For a concrete example, if x = 101 then x = 010.
Consider the following variation of the SAT problem. As input, you receive a boolean formula F on variables x1, x2, . . . , xn (note that F is not necessarily a CNF formula). The goal is to decide if F has a satisfying assignment x ∈ {0, 1}ⁿ such that x is also a satisfying assignment. Prove that this variant of SAT is NP-Complete.