<s>[INST] 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.
 [/INST] Hint
\begin{align}
a-b&= \frac1c-\frac 1b \\ &=\frac{b-c}{bc}\\ &=\frac{c-a}{abc^2} \\& =\frac{a-b}{a^2b^2c^2}.
\end{align}
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