<s>[INST] What are the possible real values of $\frac{1}{x} + \frac{1}{y}$ given $x^3 +y^3 +3x^2y^2 = x^3y^3$? Let $x^3 +y^3 +3x^2y^2 = x^3y^3$ for $x$ and $y$ real numbers different from $0$.
Then determine all possible values of $\frac{1}{x} + \frac{1}{y}$
I tried to factor this polynomial but there's no a clear factors
 [/INST] HINT Divide by $x^3y^3$ on both sides. You will get a new equation in terms of $1/x$ and $1/y.$ Can you finish from here?
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