This paper considers the computational power of anonymous message passing algorithms (henceforth, anonymous algorithms), i.e., distributed algorithms operating in a network of unidentified nodes. We prove that every problem that can be solved (and verified) by a randomized anonymous algorithm can also be solved by a deterministic anonymous algorithm provided that the latter is equipped with a 2-hop coloring of the input graph. Since the problem of 2-hop coloring a given graph (i.e., ensuring that two nodes with distance at most 2 have different colors) can by itself be solved by a randomized anonymous algorithm, it follows that with the exception of a few mock cases, the execution of every randomized anonymous algorithm can be decoupled into a generic preprocessing randomized stage that computes a 2-hop coloring, followed by a problem-specific deterministic stage. The main ingredient of our proof is a novel simulation method that relies on some surprising connections between 2-hop colorings and an extensively used graph lifting technique.