I will give an overview of the proof of a new construction of compact $G_2$ manifolds (joint work with Dominic Joyce). We resolve ($X^6$ X $S^1$)/$Z_2$ by glueing in a family of Eguchi-Hanson spaces parametrized by the singular set, two copies of a special Lagrangian submanifold $L^3$ in $X^6$. There are two key differences from the previous glueing constructions of Joyce and Kovalev/CHNP. First, there are three pieces being glued together rather than two, and second, two of the three pieces do not admit torsion-free $G_2$ structures to start with, so we need to work harder to construct a closed $G_2$ structure with sufficiently small torsion on the resolved space in order to apply Joyce's fundamental existence theorem. I plan to explain all of the main ideas and to give a few of the details of each of the principal steps in the proof.