Maucher [@guttmann06 Lemma 1], their construction is as follows. Let $I$ be an instance of [2P2N-3-SAT]{} that is given by a set of variables $U = \{u_1,\,\ldots,\,u_n\}$ and a set of clauses $${{\mathcal C}}=\{(c_{1,1} \vee c_{1,2} \vee c_{1,3}),\,\ldots,\,(c_{m,1} \vee c_{m,2} \vee c_{m,3})\},$$ where each $c_{i,j} \in \{u_1, \bar{u}_1,u_2, \bar{u}_2,\ldots,u_n, \bar{u}_n\}$. Furthermore, for $a,b\in\mathbb{N}$, let $a \oplus b$ denote the number $c \in \{1,\,2,\,3\}$ such that $a + b \equiv c \pmod 3$. We define the following three sets: $$\begin{aligned}
A &= \{u_{k,l},\,\bar{u}_{k,l} \mid 1 \leq k \leq n \wedge 1 \leq l \leq 3\} \cup {}\\
  &\hphantom{{} = {}}\{c_{i,j}^l \mid 1 \leq i \leq m \wedge 1 \leq j \leq 3 \wedge 1 \leq l \leq 3\},\\[5pt]
B &= \{(u_{k,1},\,\bar{u}_{k,3}),\,(\bar{u}_{k,1},\,u_{k,3}) \mid 1 \leq k \leq n\} \cup {}\\
  &\hphantom{{} = {}}\{(c_{i,j,2},\,c_{i,j}^{1}),\,(c_{i,j}^2,\,c_{i,j,1}) \mid 1 \leq i \leq m \wedge 1 \leq j \leq 3\} \cup {}\\
&\hphantom{{} = {}}\{(c_{i,j \oplus 1}^1,\,c_{i,j}^3) \mid 1 \leq i \leq m \wedge 1 \leq j \leq 3\},\\[5pt]
C &= \{(u_{k,1},\,u_{k,2},\,u_{k,3}),\,(\bar{u}_{k,1},\,\bar{u}_{k,2},\,\bar{u}_{k,3}) \mid 1 \leq k \leq n\} \cup {}\\
  &\hphantom{{} = {}}\{(c_{i,j}^1,\,c_{i,j}^2,\,c_{i,j}^3) \mid 1 \leq i \leq m \wedge 1 \leq j \leq 3\},\end{aligned}$$ where $c_{i,j,l}$ is an abbreviation of $u_{k,l}$ with $u_k = c_{i,j}$. By 