The measurement of the K$^-$ multi-nucleon absorptions branching ratios and low-energy cross sections in both the $\Lambda$p and $\Sigma^0$p channels are performed by the AMADEUS collaboration, exploiting the low-momentum K$^-$ (p$_K \sim$ 127 MeV/c) beam produced at the DA$\Phi$NE collider. The KLOE 2004-2005 data are analyzed by reconstructing the $\Lambda$p final state produced by the K$^-$ interactions with the inner wall of the KLOE drift chamber, which is an almost pure carbon target. Such measurements are fundamental to investigate the in-medium modification of the K$^-$ potential, which is attractive in-medium due to the partial restoration of chiral symmetry and whose behaviour in the KbarN subthreshold region is theoretically debated. Possible existence of Kbar-multinucleon bound state, whose properties are related to the controversial $\Lambda$(1405) nature, is also experimentally debated. In kaon induced reaction the exotic state formation overlaps with the multi-absorption processes over a broad range of phase space rendering their measurements mandatory. In this work the yields of the K$^-$ two and three nucleon absorptions (2NA and 3NA) are measured with unprecedented accuracy. The signal emitted by the intermediate formation of the exotic K$^-$pp bound state, decaying through the $\Lambda$p channel, is also critically investigated.