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Utility-scale battery energy storage system (BESS) technologies have huge potential to support system frequency in low-inertia conditions via fast frequency response (FFR) as well as system voltage via dynamic reactive power response. However, technical challenges may emerge in weak grids where low system strength could cause voltage instability, eventually potentially hindering provision of FFR too. This paper provides a systematic and comprehensive analysis of the challenges, and proposes relevant solutions, for BESS in providing FFR and general dynamic active and reactive responses under low system strength conditions. In particular, first we propose a constraint to describe the negative impact of low system strength on BESS phase-locked loop (PLL) stability and possible mitigation solutions. We then investigate the negative impact of large contingencies on BESS DC-link stability and active power control by introducing d-q current-power sensitivity factors and propose a solution based on reactive power prioritization that prevents converter-level instabilities while providing simultaneous active and reactive dynamic responses. Possible negative active/reactive power interactions due to BESS converter capacity constraints and the impact of the BESS operating point are also addressed. All these issues are investigated, and the proposed solutions tested, through explicit mathematical modelling as well as dynamic simulations of the South Australia's Eyre Peninsula weak grid. [ABSTRACT FROM AUTHOR]