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The ability to control and manipulate magnetic anisotropy in the colossal magnetoresistive (CMR) oxide (La,Sr)MnO3 (LSMO) is critical for its implementation in magnetic memory applications. In this work, we employ the planar Hall effect (PHE) as a powerful tool to probe the magnetic anisotropy in LSMO thin films and nanostructures, where the magnetization is too small to be detected by conventional magnetometry techniques. By analyzing the angular- and magnetic field-dependences of the PHE, we deduced an in-plane biaxial magnetocrystalline anisotropy (MCA) energy of ~1.2x10^5 erg/cm^2 in LSMO thin films fully strained on (001) SrTiO3 substrates. Creating nanoscale periodic depth modulation in LSMO establishes a uniaxial anisotropy with substantially enhanced MCA energy density, which is attributed to a high strain gradient sustained in the nanostructure. The energy competition between the biaxial and uniaxial MCA leads to multi-level resistance switching behavior in properly engineered LSMO nanostructures, which can be utilized to design the switching dynamics in magnetic memory devices. Our work points to the critical role of epitaxial strain in determining the MCA in CMR oxides, and provides an effective material strategy for engineering the magnetic properties of LSMO for novel spintronic applications with high thermal stability and high density data storage.