Floer homotopy theory and degenerate Lagrangian intersections

We give a lower bound on the number of intersection points of a Lagrangian pair via Steenrod squares on Lagrangian Floer cohomology induced from a Floer homotopy type. The main technical input is a computation of the associated graded of the action-filtration of the Floer homotopy type in terms of Morse homotopy theory (precisely, Conley index theory). We also prove a lower bound using the quantum cap product on Lagrangian Floer cohomology.