Composed Dehn twist exact sequence through A infinity n-modules

We prove the quilted Floer cochain complexes form A infinity n-modules over the Fukaya category of Lagrangian correspondences. Then we prove that when we restrict the input to mapping cones of product Lagrangians and graphs, the resulting bar-type complex can be identified with bar complex from ordinary Floer theory. As an application we use a family version of quilt unfolding argument to prove two long exact sequences conjectured by Seidel that relates the Lagrangian Floer cohomology of a collection of (possibly intersecting) Lagrangian spheres and the fixed point Floer cohomology of composition of Dehn twists along them.