= Solution
The extra ancestor multiplies the <probability generating function> by $s$. The preceding <probability generating function of an independent random sum> formula therefore yields
$$
G_Z(s)=sG_N(G_Z(s))=\frac{s}{2-G_Z(s)},\qquad G_Z(s)^2-2G_Z(s)+s=0.
$$
For $0\le s<1$ the two roots are $1\pm\sqrt{1-s}$. A <probability generating function> lies in $[0,1]$ there, so the plus root is impossible. Also $Z\ge1$ gives $G_Z(0)=0$. Consequently the <total progeny with critical geometric offspring> has
$$
\boxed{G_Z(s)=1-\sqrt{1-s},\qquad 0\le s\le1.}
$$
At $s=1$ the value is $1$, consistent with almost-sure finiteness. For complex $|s|<1$, the same expression uses the analytic square root that equals $1$ at $s=0$.
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