 Solution

The extra ancestor multiplies the probability generating function by $s$. The preceding probability generating function of an independent random sum formula therefore yieldsFor $0\le s<1$ 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 $Z\ge 1$ gives $G_Z(0)=0$. Consequently the total progeny with critical geometric offspring hasAt $s=1$ 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 $s=0$.