  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 BCG's
systemic velocity observed with ALMA and may be fueling cold accretion onto the
central black hole.
