Let $V$ be a set of $n$ vertices, and $a_n$ denote the number of undirected simple graphs $G=(V,E)$ that we can find such that $G$ and its complement $\frac{?}{\text{b}\text{a}\text{r}}(G)$ are both bipartite.

For instance, $a_1=1,a_2=2,a_3=6.$

$(15\%)$ What is the value of $a_4?$ Justify your answer.