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Various signal processing applications can be expressed as large-scale optimization problems with a composite objective structure, where the Lipschitz constant of the smooth part gradient is either not known, or its local values may only be a fraction of the global value. The smooth part may be strongly convex as well. The algorithms capable of addressing this problem class in its entirety are black-box accelerated first-order methods, related to either Nesterov's Fast Gradient Method or the Accelerated Multistep Gradient Scheme, which were developed and analyzed using the estimate sequence mathematical framework. In this paper, we develop the augmented estimate sequence framework, a relaxation of the estimate sequence. When the lower bounds incorporated in the augmented estimate functions are hyperplanes or parabolae, this framework generates a conceptually simple gap sequence. We use this gap sequence to construct the Accelerated Composite Gradient Method (ACGM), a versatile first-order scheme applicable to any composite problem. Moreover, ACGM is endowed with an efficient dynamic Lipschitz constant estimation (line-search) procedure. We also introduce the wall-clock time unit (WTU), a complexity measure applicable to all first-order methods that accounts for variations in per-iteration complexity and more consistently reflects the running time in practical applications. When analyzed using WTU, ACGM has the best provable convergence rate on the composite problem class, both in the strongly and non-strongly convex cases. Our simulation results confirm the theoretical findings and show the superior performance of our new method.