We present new, deep (245 ks) Chandra observations of the galaxy cluster Abell 1664 ($z = 0.1283$). These images reveal rich structure, including elongation and accompanying compressions of the X-ray isophotes in the NE-SW direction, suggesting that the hot gas is sloshing in the gravitational potential. This sloshing has resulted in cold fronts, at distances of 55, 115 and 320 kpc from the cluster center. Our results indicate that the core of A1664 is highly disturbed, as the global metallicity and cooling time flatten at small radii, implying mixing on large scales. The central AGN appears to have recently undergone a mechanical outburst, as evidenced by our detection of cavities. These cavities are the X-ray manifestations of radio bubbles inflated by the AGN, and may explain the motion of cold molecular CO clouds previously observed with ALMA. The estimated mechanical power of the AGN, using the minimum energy required to inflate the cavities as a proxy, is $P_{rm cav} = (1.1 pm 1.0) times 10^{44} $ erg s$^{-1}$, which may be enough to drive the molecular gas flows, and offset the cooling luminosity of the ICM, at $L_{rm cool} = (1.90 pm0.01)times 10^{44}$ erg s$^{-1}$. This mechanical power is orders of magnitude higher than the measured upper limit on the X-ray luminosity of the central AGN, suggesting that its black hole may be extremely massive and/or radiatively inefficient. We map temperature variations on the same spatial scale as the molecular gas, and find that the most rapidly cooling gas is mostly coincident with the molecular gas reservoir centered on the BCGs systemic velocity observed with ALMA and may be fueling cold accretion onto the central black hole.
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