Hyperelliptic curve residual divisors

Let $X$ be the smooth projective complex curve obtained by completing the affine curve $y^2=x^{33}-x$ over $\mathbb{C}$. Let $O$ be its unique point at infinity. For each integer $i$ with $2\le i\le 9$, fix one square root $a_i\in\mathbb{C}$ of $i^{33}-i$, and define two labeled points $P_i=(i,a_i)$ and $Q_i=(i,-a_i)$ on $X$. Let $A=\{O\}\cup\{P_i,Q_i:2\le i\le 9\}$. Divisors supported on $A$ are integral linear combinations of these $17$ labeled points, and an effective divisor has all coefficients nonnegative.

Let $K$ be the divisor $30O$. For any divisor $D$ on $X$, write $\ell(D)=\dim_{\mathbb{C}}H^0(X,\mathcal{O}_X(D))$; this convention also applies when $D$ has negative coefficients. Let $S$ be the set of effective divisors $E$ supported on $A$ with degree $16$ such that $\ell(E)>1$ and satisfying the following condition. There must exist an effective divisor $F$ supported on $A$ with degree $14$ such that $E+F$ is linearly equivalent to $K$.

For such an $F$, let $\operatorname{Supp}(F)$ be the set of points of $A$ with positive coefficient in $F$, not counted with multiplicity. Let $N_F$ be the number of unordered pairs $\{r,s\}$, with $r,s\in\operatorname{Supp}(F)$ and repetitions allowed, such that $\ell(F+r+s)>1$; when $r=s$, the divisor $F+r+s$ means $F+2r$. The required condition is that at least one such $F$ has $N_F>0$ and $N_F$ congruent modulo $2$ to $|\operatorname{Supp}(F)|+\ell(K-E)$.

Let $\tau:X\to X$ be the involution $\tau(x,y)=(x,-y)$, extended by $\tau(O)=O$. It acts on divisors supported on $A$ by $\tau(P_i)=Q_i$, $\tau(Q_i)=P_i$, and $\tau(O)=O$, preserving coefficients. Two divisors in $S$ are in the same $\tau$-orbit if and only if one is equal to the image of the other under this action; the indices $i=2,\ldots,9$ are not permuted. A divisor $E$ in $S$ is $\tau$-fixed if $\tau(E)=E$ as a labeled divisor.

Determine the ordered pair whose first entry is the number of $\tau$-orbits in $S$ and whose second entry is the number of $\tau$-fixed divisors in $S$.