  For every $g \gg 1$, we show the existence of a complete and smooth family of
closed constant mean curvature surfaces $f_\varphi^g,$ $ \varphi \in [0,
\tfrac{\pi}{2}],$ in the round $3$-sphere deforming the Lawson surface $\xi_{1,
g}$ to a doubly covered geodesic 2-sphere with monotonically increasing
Willmore energy. To construct these we use an implicit function theorem
argument in the parameter $t= \tfrac{1}{2(g+1)}$. This allows us to give an
iterative algorithm to compute the power series expansion of the DPW potential
and area of $f_\varphi^g$ at $t= 0$ explicitly. In particular, we obtain for
large genus Lawson surfaces $\xi_{1,g}$ % due to the real analytic dependence
of its area and DPW potential on $t,$ a scheme to explicitly compute the
coefficients of the power series in $t$ in terms of multiple polylogarithms.
Remarkably, the third order coefficient of the area expansion is identified
with $\tfrac{9}{4}\zeta(3),$ where $\zeta$ is the Riemann $\zeta$ function
(while the first and second order term were shown to be $\log(2)$ and $0$
respectively in \cite{HHT}).
