= Solution
Parametrize the <torus> by $F(u,v)=((2+\cos u)\cos v,(2+\cos u)\sin v,\sin u)$. Its outward <Gauss map> is
$$
N(u,v)=(\cos u\cos v,\cos u\sin v,\sin u).
$$
The derivative $N_u$ has unit length, while $N_v$ has length $|\cos u|$ and is perpendicular to $N_u$. Thus rank drops precisely at $u=\pi/2$ or $3\pi/2$. These are the two circles of height $z=1$ and $z=-1$ and radius two, and their images are the north and south poles. Therefore \b[the critical values are $(0,0,\pm1)$].
For any nonpolar target point, its third coordinate determines two values of $u$, one with positive and one with negative $\cos u$. For each, the longitude fixes a unique $v$ modulo $2\pi$, with a shift of $\pi$ when $\cos u$ changes sign. Thus every <regular value> has exactly two preimages, one on the outer and one on the inner side of the <torus>. Consequently
$$
\boxed{\deg_2N=0.}
$$
This explicit two-sheet description justifies the count without relying on a sketch.
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