  The fascinating realm of non-Hermitian physics with the interplay of parity
(P) and time-reversal (T) symmetry has been witnessing immense attention in
exploring unconventional physics at Exceptional Point (EP) singularities.
Particularly, the physics of PT-symmetry, anti-PT (APT)-symmetry, and the
emergence of EPs have ignited fervor in photonics. Beyond the conventional
relation between EP and PT-symmetric phase transitions, this study delves into
hosting higher-order EPs in a specially designed APT-symmetric
Fabry-P\'erot-type microcavity. We unveil the captivating physics of the
parametric encirclement schemes to explore the branch-point behaviors of EPs up
to order three in terms of successive state-flipping, while optimizing the
designed cavity under APT-symmetric constraints. The insights from our findings
are poised to boost research in optical metamaterials, meeting the demands of
APT-symmetry and paving the way for a novel class of photonic devices.
