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The nucleosynthetic site of the rapid ($r$) neutron-capture process is currently being debated. The direct detection of the neutron star merger GW170817, through gravitational waves and electromagnetic radiation, has confirmed such events as important sources of the $r$-process elements. However, chemical evolution models are not able to reproduce the observed chemical abundances in the Milky Way when neutron star mergers are assumed to be the only $r$-process site and realistic time distributions of such events are taken into account. Now for the first time, we combine all the available observational evidence of the Milky Way and its dwarf galaxy satellites to show that the data can only be explained if there are (at least) two distinct $r$-process sites: a quick source with timescales comparable to core-collapse supernovae, $t_{quick}\lesssim10^8$ yr, and a delayed source with characteristic timescales $t_{delayed}\gtrsim4$ Gyr. The delayed $r$-process source most probably originates in neutron star mergers, as the timescale fits well with that estimated for GW170817. Given the short timescales of the quick source, it is likely associated with massive stars, though a specific fast-track channel for compact object mergers cannot be excluded at this point. Our approach demonstrates that only by looking at all the available data will we be able to solve the puzzle that is the $r$-process.