Each $f$ lies in $zA$ for some $A$.
Note that some of the $X_n$ may be repeated.
Some of the isomorphism classes above will have a rank of 2.
Some such difficulty is to be expected.
The theorem implies that some finite subcollection of the $f_i$ can be removed without altering the span.
It is therefore reasonable that the behaviour of $p$ should in some rough sense approximate the behaviour of $q$.