We present an analysis of deep Chandra X-ray observations of the galaxy cluster MS 0735.6+7421, which hosts the most energetic radio AGN known. Our analysis has revealed two cavities in its hot atmosphere with diameters of 200-240 kpc. The total cavity enthalpy, mean age, and mean jet power are $9times 10^{61}$ erg, $1.6times 10^{8}$ yr, and $1.7times 10^{46}$ erg/s, respectively. The cavities are surrounded by nearly continuous temperature and surface brightness discontinuities associated with an elliptical shock front of Mach number 1.26 (1.17-1.30) and age of $1.1times 10^{8}$ yr. The shock has injected at least $4times 10^{61}$ erg into the hot atmosphere at a rate of $1.1times 10^{46}$ erg/s. A second pair of cavities and possibly a second shock front are located along the radio jets, indicating that the AGN power has declined by a factor of 30 over the past 100 Myr. The multiphase atmosphere surrounding the central galaxy is cooling at a rate of 36 Msun/yr, but does not fuel star formation at an appreciable rate. In addition to heating, entrainment in the radio jet may be depleting the nucleus of fuel and preventing gas from condensing out of the intracluster medium. Finally, we examine the mean time intervals between AGN outbursts in systems with multiple generations of X-ray cavities. We find that, like MS0735, their AGN rejuvenate on a timescale that is approximately 1/3 of their mean central cooling timescales, indicating that jet heating is outpacing cooling in these systems.
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