We report on the decay of the $f_1(1285)$ into $\pi a_0(980)$ and $K^* \bar K$. We find a shoulder around 1400 MeV, tied to a triangle singularity, for the $\pi a_0(980)$ decay mode, and a peak around 1420 MeV for the $K^* \bar K$ mode. Both these features agree with the experimental information on which the $f_1(1420)$ resonance is based. Together with other features we conclude that the $f_1(1420)$ is not a genuine resonance, but the manifestation of the $\pi a_0(980)$ and $K^* \bar K$ decay modes of the $f_1(1285)$ at higher energies than the nominal one.