In conformal field theory, the lightcone bootstrap is an analytic approach to solving the crossing equations of correlation functions. The blocks, namely the kinematical constituents of a crossing equation, must be computed near lightcone singularities and resumed. At four points, this culminates in the universal dynamics of a double-twist operator at large spin. While extensions to higher points have been recently studied, further progress has been hampered by our limited knowledge of the blocks. In this talk, I will review the first applications of conformal block integrability to the multipoint lightcone bootstrap, focusing on scalar five point functions for concrete expressions. First, starting from a Gaudin model, I will review the construction of an integrable system that determines blocks via differential equations and a boundary condition. These differential equations, and by extension the crossing equations, can be explicitly solved in lightcone limits. Finally, after summarizing the old and new results we obtain at five points, I will comment on what this may entail for universal triple-twist data in six point functions. This is based on my thesis work, as well as an upcoming paper with Lorenzo Quintavalle, Apratim Kaviraj and Volker Schomerus.