Composed Dehn twist exact sequence through quilts and (A∞,n) modules

We prove the quilted Floer cochain complexes form $(A_\infty,n)$ modules over the Fukaya category $Fuk(M \times M^-)$. 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 prove two long exact sequences conjectured by Seidel that relates the Lagrangian Floer cohomology of a collection of (possibly intersecting) Lagrangian spheres $L_i$ and the fixed point Floer cohomology of composed Dehn twists $\tau_{L_1} o ...  \tau_{L_n}$ along them.