To understand the role that AGN feedback plays in galaxy evolution we need in-depth studies of the multi-phase structure and energetics of galaxy-wide outflows. In this work we present new, deep ($sim$50 hr) NOEMA CO(1-0) line observations of the molecular gas in the powerful outflow driven by the AGN in the ultra-luminous infrared galaxy IRAS F08572+3915. We spatially resolve the outflow, finding that its most likely configuration is a wide-angle bicone aligned with the kinematic major axis of the rotation disk. The molecular gas in the wind reaches velocities up to approximately $pm$1200 km s$^{-1}$ and transports nearly 20% of the molecular gas mass in the system. We detect a second outflow component located $sim$6 kpc north-west from the galaxy moving away at $sim$900 km s$^{-1}$, which could be the result of a previous episode of AGN activity. The total mass and energetics of the outflow, which includes contributions from the ionized, neutral, warm and cold molecular gas phases is strongly dominated by the cold molecular gas. In fact, the molecular mass outflow rate is higher than the star formation rate, even if we only consider the gas in the outflow that is fast enough to escape the galaxy, which accounts for about $sim$40% of the total mass of the outflow. This results in an outflow depletion time for the molecular gas in the central $sim$1.5 kpc region of only $sim3$ Myr, a factor of $sim2$ shorter than the depletion time by star formation activity.
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