The NA61/SHINE experiment at the CERN Super Proton Synchrotron (SPS) studies the onset of deconfinement in hadron matter by a scan of particle production in collisions of nuclei with various sizes at a set of energies covering the SPS energy range. This paper presents results on inclusive double-differential spectra, transverse momentum and rapidity distributions and mean multiplicities of $\pi^\pm$, $K^\pm$, $p$ and $\bar{p}$ produced in the 20$\%$ most $central$$^7$Be+$^9$Be collisions at beam momenta of 19$A$, 30$A$, 40$A$, 75$A$ and 150$A$ GeV/$c$. The energy dependence of the $K^\pm$/$\pi^\pm$ ratios as well as of inverse slope parameters of the $K^\pm$ transverse mass distributions are close to those found in inelastic $p$+$p$ reactions. The new results are compared to the world data on $p$+$p$ and Pb+Pb collisions as well as to predictions of the EPOS, UrQMD, AMPT, PHSD and SMASH models.
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