Answer set programming (ASP) is a well-established logic programming language that offers an intuitive, declarative syntax for problem solving. In its traditional application, a fixed ASP program for a given problem is designed and the actual instance of the problem is fed into the program as a set of facts. This approach typically results in programs with comparably short and simple rules. However, as is known from complexity analysis, such an approach limits the expressive power of ASP; in fact, an entire NP-check can be encoded into a single large rule body of bounded arity that performs both a guess and a check within the same rule. Here, we propose a novel paradigm for encoding hard problems in ASP by making explicit use of large rules which depend on the actual instance of the problem. We illustrate how this new encoding paradigm can be used, providing examples of problems from the first, second, and even third level of the polynomial hierarchy. As state-of-the-art solvers are tuned towards short rules, rule decomposition is a key technique in the practical realization of our approach. We also provide some preliminary benchmarks which indicate that giving up the convenient way of specifying a fixed program can lead to a significant speed-up.