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We report new ALMA observations of the CO(3-2) line emission from the 2.1 ± 0.3 × 1010 M⊙ molecular gas reservoir in the central galaxy of the Phoenix cluster. The cold molecular gas is fueling a vigorous starburst at a rate of 500-800 M⊙ yr-1 and powerful black hole activity in the forms of both intense quasar radiation and radio jets. The radio jets have inflated huge bubbles filled with relativistic plasma into the hot, X-ray atmospheres surrounding the host galaxy. The ALMA observations show that extended filaments of molecular gas, each 10-20 kpc long with a mass of several billion solar masses, are located along the peripheries of the radio bubbles. The smooth velocity gradients and narrow line widths along each filament reveal massive, ordered molecular gas flows around each bubble, which are inconsistent with gravitational free-fall. The molecular clouds have been lifted directly by the radio bubbles, or formed via thermal instabilities induced in low-entropy gas lifted in the updraft of the bubbles. These new data provide compelling evidence for close coupling between the radio bubbles and the cold gas, which is essential to explain the self-regulation of feedback. The very feedback mechanism that heats hot atmospheres and suppresses star formation may also paradoxically stimulate production of the cold gas required to sustain feedback in massive galaxies.