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A new methodology, advanced clustering with frequency sweeping (ACFS), is introduced for the stability analysis of the most general class of linear time-invariant time-delay systems with multiple delays. ACFS is a cross fertilization of algebraic geometry, frequency sweeping technique and the root clustering paradigm, and it can directly extract the 2-D cross sections of the stability views in any two-delay domain. In this domain, ACFS can also test the delay-independent stability based on necessary and sufficient conditions, and can compute, for the delay-dependent case, the precise lower and upper bounds of the only parameter, the frequency, that ACFS sweeps. Case studies that are prohibitive to analyze with the existing analytical methods are presented to demonstrate the strengths of the method.