$$C \to A B C \mid A$$
Obviously has the same FIRST(C) for the both productions.
But how can this be left-factorised?
$$C \to A B C'$$ $$C' \to A C \mid \varepsilon$$
The FIRST sets are disjoint, but is this correct?
Should $B \to \varepsilon$ (or $C \to \varepsilon$) be added as well, so that it handles $A B A$?