The neutron and its hypothetical mirror counterpart, a sterile state degenerate in mass, could spontaneously mix in a process much faster than the neutron $beta$-decay. Two groups have performed a series of experiments in search of neutron - mirror-neutron ($n-n$) oscillations. They reported no evidence, thereby setting stringent limits on the oscillation time $tau_{nn}$. Later, these data sets have been further analyzed by Berezhiani et al.(2009-2017), and signals, compatible with $n-n$ oscillations in the presence of mirror magnetic fields, have been reported. The Neutron Electric Dipole Moment Collaboration based at the Paul Scherrer Institute performed a new series of experiments to further test these signals. In this paper, we describe and motivate our choice of run configurations with an optimal filling time of $29~$s, storage times of $180~$s and $380~$s, and applied magnetic fields of $10~mu$T and $20~mu$T. The choice of these run configurations ensures a reliable overlap in settings with the previous efforts and also improves the sensitivity to test the signals. We also elaborate on the technique of normalizing the neutron counts, making such a counting experiment at the ultra-cold neutron source at the Paul Scherrer Institute possible. Furthermore, the magnetic field characterization to meet the requirements of this $n-n$ oscillation search is demonstrated. Finally, we show that this effort has a statistical sensitivity comparable to the current leading constraints for $n-n$ oscillations.
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