a,b,c are three real numbers where $a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}$. Now $abc$ = ? Here will the answer be a number? I want to know whether it is possible to get a real number (not an algebraic expression)  as the product of $a$, $b$ and $c$.
I tried for a long time and this is what I got.
$$3abc = a^2 b + b^2 c + c^2 a$$
But it seems like there is a way to determine the value.

If $a=b$ then $b=c$ and $a=b=c$, which says that $abc$ is not defined. 
Let $(a-b)(a-c)(b-c)\neq0.$
Thus, since 
$$a-b=\frac{1}{c}-\frac{1}{b}=\frac{b-c}{bc},$$
$$b-c=\frac{c-a}{ac}$$ and
$$c-a=\frac{a-b}{ab},$$ we obtain
$$(a-b)(b-c)(c-a)=\frac{(a-b)(b-c)(c-a)}{a^2b^2c^2},$$ which gives
$$abc=1$$ or $$abc=-1.$$
