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Ethanol (CH$_3$CH$_2$OH) is a relatively common molecule, often found in star forming regions. Recent studies suggest that it could be a parent molecule of several so-called interstellar complex organic molecules (iCOMs). Yet, the formation route of this species remains debated. In the present work, we study the formation of ethanol through the reaction of CCH with one H$_2$O molecule belonging to the ice, as a test case to investigate the viability of chemical reactions based on a "radical + ice component" scheme as an alternative mechanism for the synthesis of iCOMs, beyond the usual radical-radical coupling. This has been done by means of DFT calculations adopting two clusters of 18 and 33 water molecules as ice models. Results indicate that CH$_3$CH$_2$OH can potentially be formed by this proposed reaction mechanism. The reaction of CCH with H$_2$O on the water ice clusters can be barrierless (thanks to the help of boundary icy water molecules acting as proton transfer assistants) leading to the formation of vinyl alcohol precursors (H$_2$CCOH and CHCHOH). Subsequent hydrogenation of vinyl alcohol yielding ethanol is the only step presenting a low activation energy barrier. We finally discuss the astrophysical implications of these findings., 30 pages, 6 figures Accepted for publication ACS Earth and Space Chemistry