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Galaxy clusters and cosmic voids, the most extreme objects of our Universe in terms of mass and size, trace two opposite sides of the large-scale matter density field. By studying their abundance as a function of their mass and radius, respectively, i.e. the halo mass function (HMF) and void size function (VSF), it is possible to achieve fundamental constraints on the cosmological model. While the HMF has already been extensively exploited providing robust constraints on the main cosmological model parameters (e.g. $Ω_{\rm m}$, $σ_8$ and $S_8$), the VSF is still emerging as a viable and effective cosmological probe. Given the expected complementarity of these statistics, in this work we aim at estimating the costraining power deriving from their combination. To this end, we exploit realistic mock samples of galaxy clusters and voids extracted from state-of-the-art large hydrodynamical simulations, in the redshift range $0.2 \leq z \leq 1$. We perform an accurate calibration of the free parameters of the HMF and VSF models, needed to take into account the differences between the types of mass tracers used in this work and those considered in previous literature analyses. Then, we obtain constraints on $Ω_{\rm m}$ and $σ_8$ by performing a Bayesian analysis. We find that cluster and void counts represent powerful independent and complementary probes to test the cosmological framework. In particular, the constraining power of the HMF on $Ω_{\rm m}$ and $σ_8$ improves drastically with the VSF contribution, increasing the $S_8$ constraint precision by a factor of about 60%.