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We theoretically study the thermodynamic properties of a strongly interacting Fermi gas at the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC), by applying a recently outlined strong-coupling theory that includes pair fluctuations beyond the commonly-used many-body $T$-matrix or ladder approximation at the Gaussian level. The beyond Gaussian pair fluctuation (GPF) theory always respects the exact thermodynamic relations and recovers the Bogoliubov theory of molecules in the BEC limit with a nearly correct molecule-molecule scattering length. We show that the beyond-GPF theory predicts quantitatively accurate ground-state properties at the BEC-BCS crossover, in good agreement with the recent measurement by Horikoshi \textit{et al.} in Phys. Rev. X \textbf{7}, 041004 (2017). In the unitary limit with infinitely large $s$-wave scattering length, the beyond-GPF theory predicts a reliable universal energy equation of state up to 0.6$T_c$, where $T_c$ is the superfluid transition temperature at unitarity. The theory predicts a Bertsch parameter $ξ\simeq 0.365$ at zero temperature, in good agreement with the latest quantum Monte Carlo result $ξ= 0.367(7)$ and the latest experimental measurement $ξ= 0.367(9)$. We attribute the excellent and wide applicability of the beyond-GPF theory in the broken-symmetry phase to the reasonable re-summation of Feynman diagrams following a dimensional $ε$-expansion analysis near four dimensions ($d=4-ε$), which gives rise to accurate predictions at the second order $\mathcal{O}(ε^2)$. Our work indicates the possibility of further improving the strong-coupling theory of strongly interacting fermions based on the systematic inclusion of large-loop Feynman diagrams at higher orders $\mathcal{O}(ε^n)$ with $n\ge 3$.