  We derive a Floer theoretical SYZ mirror for an equilateral and generic
polygon space. The disk potential function of the monotone torus fiber of the
caterpillar bending system is calculated by computing non-trivial open
Gromov--Witten invariants from the structural result of the monotone Fukaya
category, the topology of fibers of completely integrable systems, and toric
degenerations. Then, combining the result with the work of Nohara--Ueda [NU20]
and Marsh--Rietsch [MR20], we obtain the disk potential functions of bending
systems and produce a mirror cluster variety of type A without frozen variables
via Lagrangian Floer theory.
