jmc

Since $f(-1)=\frac{a}{1-(-1)}=\frac{a}{2}$, we can simplify the second expression to $$\frac{a}{2}=f^{-1}(4a+1).$$This is equivalent to $$f\left(\frac{a}{2}\right)=4a+1.$$Since we know $f$, we can evaluate this as $$\frac{a}{1-\frac{a}{2}}=4a+1.$$or $$\frac{2a}{2-a}=4a+1.$$Assuming $a\ne 2$, cross multiplication gives $$2a=(4a+1)(2-a)=-4a^2+7a+2,$$so $4a^2-5a-2=0$. We note that $a=2$ is not a solution to this equation. By Vieta's formulas, the product of the roots of the quadratic equation $a{x}^2+bx+c=0$ is $c/a$, so in this case, the product of the roots is $-2/4=\boxed{-\frac{1}{2}}$.