obtained by removing invariant disks around pairs of points in two distinct orbits of the $2\pi/5$ rotation of $S_0$, and then attaching five 1-handles with full twists. This example indicates that a classification of such roots would require the examination of the orbit information of finite cyclic actions, in addition to their fixed point data. This is a significant departure from existing theories developed in [@MK1] and [@KR1].

Any subset of a multicurve will be called a *submulticurve*. A multicurve ${\operatorname{{\mathcal{C}}}}$ in $S_g$ is said to be *pseudo-nonseparating* if ${\operatorname{{\mathcal{C}}}}$ separates $S_g$, but no proper submulticurve of ${\operatorname{{\mathcal{C}}}}$ separates $S_g$. A multicurve that contains no pseudo-nonseparating submulticurves will be called a *nonseparating multicurve*, while a multicurve which is a disjoint union of pseudo-nonseparating multicurves will be called a *separating multicurve*. A multicurve that is neither separating nor nonseparating will be called a *mixed multicurve*. In Figure \[fig:s5mixed\] below, the collection of curves ${\operatorname{{\mathcal{C}}}}= \{c_1,c_2,c_3,c_4\}$ is a mixed multicurve, while the subcollections $\{c_2,c_3\}$, $\{c_1,c_2,c_3\}$ and $\{c_2,c_4\}$ form pseudo-nonseparating, separating and nonseparating multicurves, respectively.

$c_1$ at 205 120 $c_2$ at 565 160 $c_3$ at 565 85 $c_4$ at 1053 85

We start by generalizing the notion of a nestled $(n,\ell)$-action from [@KR1] 