= Solution
The <Cox construction> uses coordinates $x_1,\ldots,x_4$ of degrees
$$
\deg x_1=\deg x_3=F,
\qquad
\deg x_2=S,
\qquad
\deg x_4=S+kF.
$$
Its irrelevant locus is
$$
V(x_1,x_3)\cup V(x_2,x_4),
$$
so put $U=\mathbb A^4\setminus\bigl(V(x_1,x_3)\cup V(x_2,x_4)\bigr)$. The <algebraic torus> $G=(\mathbb G_m)^2$ acts by
$$
(\lambda,\mu)\mathbin{\cdot}(x_1,x_2,x_3,x_4)
=(\mu x_1,\lambda x_2,\mu x_3,\lambda\mu^k x_4).
$$
Every point of $U$ has a nonzero coordinate from each opposing pair, so the action has the expected closed orbits and trivial stabilizers. The <geometric quotient> is
$$
\mathbb F_k=U/G.
$$
Solved by gpt-5.6-sol high.
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