(Abridged) The relaxed cool-core Phoenix cluster (SPT-CL J2344-4243) features an extremely strong cooling flow, as well as a mini-halo. Strong star-formation in the brightest cluster galaxy indicates that AGN feedback has been unable to inhibit this cooling flow. We have studied the strong cooling flow in the Phoenix cluster by determining the radio properties of the AGN and its lobes. In addition, we use spatially resolved observations to investigate the origin of the mini-halo. We present new Very Large Array 1-12 GHz observations of the Phoenix cluster which resolve the AGN and its lobes in all four frequency bands, and resolve the mini-halo in L- and S-band. Using our L-band observations, we measure the total flux density of the radio lobes at 1.5 GHz to be $7.6pm0.8$ mJy, and the flux density of the mini-halo to be $8.5pm0.9$ mJy. Using L- and X-band images, we produce the first spectral index maps of the lobes from the AGN and measure the spectral indices of the northern and southern lobes to be $-1.35pm0.07$ and $-1.30pm0.12$, respectively. Similarly, using L- and S-band data, we map the spectral index of the mini-halo, and obtain an integrated spectral index of $alpha=-0.95 pm 0.10$. We find that the mini-halo is most likely formed by turbulent re-acceleration powered by sloshing in the cool core due to a recent merger. In addition, we find that the feedback in the Phoenix cluster is consistent with the picture that stronger cooling flows are to be expected for massive clusters like the Phoenix cluster, as these may feature an underweight supermassive black hole due to their merging history. Strong time variability of the AGN on Myr-timescales may help explain the disconnection between the radio and the X-ray properties of the system. Finally, a small amount of jet precession likely contributes to the relatively low ICM re-heating efficiency of the mechanical feedback.
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