= Solution
Apply the same <pumping of a group multiplier automaton> to the accepted pair containing the given representative $u$. Since $|u|>|w|+N$, there is a nonempty loop entirely after the tape containing $w$ has ended. Write $u=pqr$, with $q\ne\varepsilon$ the letters read during this loop. Repeating the loop any number of times leaves the other tape equal to $w$, and the <group multiplier automaton> accepts the resulting <padded convolution of words>. Consequently
$$
\boxed{pq^kr\in L,\qquad \overline{pq^kr}=g\quad(k\geq0).}
$$
These <words over an alphabet> have distinct lengths because $|q|>0$. Hence \b[there are infinitely many representatives of $g$ in $L$]. No uniqueness of representatives was assumed in either part.
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