NL-NWO 1983 Problem 3

Suppose that $a,b,c,p$ are real numbers with $a,b,c$ not all equal, such that: $a+\frac{1}{b}=b+\frac{1}{c}=c+\frac{1}{a}=p.$ Determine all possible values of $p$ and prove that $abc+p=0$ .