A JSJ-type decomposition theorem for symplectic fillings

We establish a JSJ-type decomposition theorem for splitting exact symplectic fillings of contact 3-manifolds along \emph{mixed tori} -- these are convex tori satisfying a particular geometric condition. As an application, we show that if \((M,\xi)\) is obtained from \((S^3,\xi_{\mathrm{std}})\) via Legendrian surgery along a knot which has been stabilized both positively and negatively, then \((M,\xi)\) has a unique exact filling.