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We consider the O(n) loop model on dynamical random lattices, which was previously studied by Kostov, Eynard et al using matrix integrals. Here, we revisit the problem using a combinatorial approach. An elementary decomposition, which consists in cutting the configurations along the outermost loops, allows to relate the O(n) model to the simpler problem of counting maps with controlled face degrees. This translates into a functional relation for the ``resolvent'' of the model, which is exactly solvable in several interesting cases. We then look for critical points of the model: our construction shows that at the so-called non-generic critical points, the O(n) model is related to the ``stable'' maps introduced by Le Gall and Miermont, and we will comment on the geometrical implications of this connection.