  We present expressions for the kinematic endpoints of 5-body cascade decay
chains proceeding through all possible combinations of 2-body and 3-body
decays, with one stable invisible particle in the final decay stage. When an
invariant mass can be formed in multiple ways by choosing different final state
particles from a common vertex, we introduce techniques for finding the
sub-leading endpoints for all indistinguishable versions of the invariant mass.
In contrast to short decay chains, where sub-leading endpoints are linearly
related to the leading endpoints, we find that in 5-body decays, they provide
additional independent constraints on the mass spectrum.
