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We present the generalized quasiclassical theory of the long-range superconducting proximity effect in heterostructures with strong ferromagnets, where the exchange splitting is of the order of Fermi energy. In the ferromagnet the propagation of equal-spin Cooper pairs residing on the spin-split Fermi surfaces is shown to be governed by the spin-dependent Abelian gauge field which results either from the spin-orbital coupling or from the magnetic texture. This additional gauge field enters into the quasiclassical equations in superposition with the usual electromagnetic vector potential and results in the generation of spontaneous superconducting currents and phase shifts in various geometries which provide the sources of long-range spin-triplet correlations. We derive the Usadel equations and boundary conditions for the strong ferromagnet and consider several generic examples of the Josephson systems …