Three. (20 points) Suppose the equations $x^{2}-m x+5+m=0$ and $x^{2}-$ $(17 m+1) x+13 m+7=0$ have at least one common root. Find all possible values of the product of the four roots of these two equations.

Given the equations $x^{2}-m x+5+m=0$ and $x^{2}-$ $(17 m+1) x+13 m+7=0$ have at least one common root, we need to determine all possible values of the product of the four roots of these two equations.
Let $\alpha$ be the common root of the two equations, then
$\alpha^{2}-m \alpha+5+m=0$,
$a^{2}-(7 m+1) \alpha+13 m+7=0$.
(1)-(2) gives $(6 m+1) \alpha=12 m+2$.
If $m=-\frac{1}{6}$, the two equations are identical,

thus the required product is $\frac{841}{36}$.
If $m \neq-\frac{1}{6}$, from (3) we get $\alpha=2$, substituting into (1) gives $m=$ 9. Therefore, the required product is 1736.